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Multiplying algebraic expressions with more than one term
\latex{ (4+5)\times(40+5) }
€\latex{40}
€\latex{5}
Multiplying a two-term algebraic expression by an algebraic expression with more than one term
Example 1
Some Hungarian students took part in an exchange trip to Paris during spring break. After visiting the Eiffel Tower, the Hungarian and French students ate ice cream together. How much money did they spend on the tickets and the ice cream if
a) \latex{ four } Hungarian students and \latex{ five } French students took part in the activities, and the tickets cost \latex{ 40 } \latex{ euros } \latex{ per } \latex{ person } and the ice cream cost \latex{5} \latex{ euros } \latex{ per } \latex{ person; }
b) \latex{\large a} Hungarian students and \latex{\large b } French students took part in the activities, and the tickets cost \latex{\large c } \latex{ euros } \latex{ per } \latex{ person } and the ice cream cost \latex{\large d } \latex{ euros } \latex{ per } \latex{ person? }
b) \latex{\large a} Hungarian students and \latex{\large b } French students took part in the activities, and the tickets cost \latex{\large c } \latex{ euros } \latex{ per } \latex{ person } and the ice cream cost \latex{\large d } \latex{ euros } \latex{ per } \latex{ person? }
Solution
You can calculate in two different ways. According to the first method, first find out how much one student spent in total, then multiply that amount by the total number of students.
According to the second method, calculate how much the Hungarian students spent on tickets and on ice cream, and how much the French students spent on tickets and on ice cream, then add the two totals together.
According to the second method, calculate how much the Hungarian students spent on tickets and on ice cream, and how much the French students spent on tickets and on ice cream, then add the two totals together.
a) Using the first method: there were a total of \latex{ 4 + 5 } students, and each student spent \latex{ 40 + 5 } euros. Therefore, the total cost was
\latex{ (4 + 5) \times (40 + 5) = 9 \times 45 =} €\latex{ 405 }.
Using the second method: the Hungarian students spent \latex{ 4 \times 40 } \latex{ euros } on tickets and \latex{ 4\times5 } \latex{ euros } on ice cream, while the French students spent \latex{ 5\times40 } \latex{ euros } on tickets and \latex{ 5\times5 } \latex{ euros } on ice cream. Thus, the total cost was
\latex{ 4 \times 40 + 4 \times 5 + 5\times 40 + 5 \times5 = 160 + 20 + 200 + 25 =} €\latex{ 405 }.
You get the same result using both methods.
\latex{ (\textcolor{#00adee}{4} + \textcolor{#ed155b}{5}) \times (40 + 5) = \textcolor{#00adee}{4} \times 40 + \textcolor{#00adee}{4} \times 5 + \textcolor{#ed155b}{5} \times 40 + \textcolor{#ed155b}{5} \times 5. }
b) Make a table.
students
\latex{a+b }
costs: \latex{ c+d }
ticket: \latex{ c }
ice cream: \latex{ d }
\latex{ a \times d }
\latex{ b \times d }
\latex{ b \times c }
\latex{ a\times c }
Hungarian: \latex{ a }
French: \latex{b }
Using the first method: there were a total of \latex{ a + b } students, and each student paid\latex{ (c + d) } \latex{ euros. } Therefore, the total cost was \latex{ (a + b) \times (c + d) } \latex{ euros. }
Using the second method: the Hungarian students spent \latex{ a \times c } \latex{ euros } on tickets and \latex{ a \times d } \latex{ euros } on ice cream, while the French students spent \latex{ b \times c } \latex{ euros } on tickets and \latex{ b \times d } \latex{ euros } on ice cream. Therefore, the total cost was \latex{ a \times c + a \times d + b \times c + b \times d } \latex{ euros. }
You get the same result using both methods.
\latex{ (\textcolor{#00adee}{a} + \textcolor{#ed155b}{b}) \times (c + d) = \textcolor{#00adee}{a} \times c + \textcolor{#00adee}{a} \times d + \textcolor{#ed155b}{b} \times c + \textcolor{#ed155b}{b} \times d. }
When multiplying algebraic expressions consisting of two terms, multiply each term in the first expression by each term in the second, then add the products together.
The multiplication of algebraic expressions consisting of two terms can be illustrated using rectangles.
\latex{(a+b) \times (c+d)}
\latex{a}
\latex{b}
\latex{c}
\latex{d}
One side of the large rectangle is \latex{ a+b } units long, while the other side is \latex{ c+d } units long. Therefore, the area of the rectangle is \latex{ (a + b) \times (c + d) }.
\latex{a \times c}
\latex{b}
\latex{a}
\latex{c}
\latex{d}
\latex{a \times d}
\latex{b \times d}
\latex{b \times c}
The large rectangle consists of four smaller rectangles. The sum of their areas is \latex{ ac + ad + bc + bd }, which is equal to the area of the large rectangle.
Therefore,
\latex{ (a + b) \times (c + d) = ac + ad + bc + bd. }
The multiplication of expressions involving subtraction can be carried out in a similar way.
\latex{ \textcolor{#00adee}{(a - b) \times (c + d)} = (a + (-b)) \times (c + d) = }
\latex{ = ac + ad + (-b) \times c + (-b) \times d = \textcolor{#00adee}{ac + ad - bc - bd} } and
\latex{ \textcolor{#00adee}{(a - b) \times (c - d)} = (a + (-b)) \times (c + (-d)) = }
\latex{ = a \times c + a \times (-d) + (-b) \times c + (-b) \times (-d) = \textcolor{#00adee}{ac - ad - bc + bd} }
\latex{ = ac + ad + (-b) \times c + (-b) \times d = \textcolor{#00adee}{ac + ad - bc - bd} } and
\latex{ \textcolor{#00adee}{(a - b) \times (c - d)} = (a + (-b)) \times (c + (-d)) = }
\latex{ = a \times c + a \times (-d) + (-b) \times c + (-b) \times (-d) = \textcolor{#00adee}{ac - ad - bc + bd} }
Example 2
Perform the following multiplication of algebraic expressions:
\latex{ (x + 2)\times (a + b + 3) }
Solution
\latex{2 \times 3}
\latex{2}
\latex{x}
\latex{a}
\latex{b}
\latex{3}
\latex{x \times 3}
\latex{x \times b}
\latex{x \times a}
\latex{2 \times a}
\latex{2 \times b}
In this case as well, multiply each term in the first factor by each term in the second factor.
\latex{ (x + 2) \times (a + b + 3)=}
\latex{ = ax + bx + 3x + 2a + 2b + 2 \times 3 =}
\latex{ =xa + xb + 3x + 2a + 2b + 6 }
Example 3
Two natural numbers are divided by six. The first number leaves a remainder of one, and the second number leaves a remainder of two. What will the remainder be when the following are divided by six:
a) their sum b) their product?
Solution
Natural numbers that leave a remainder of one when divided by six can be represented by the expression \latex{ 6a + 1 }, where \latex{ a } is a natural number. Numbers that leave a remainder of two can be represented by \latex{ 6b + 2 }, where \latex{ b } is a natural number.
a) The sum of the two numbers.
\latex{ 6a + 1 + 6b + 2 = \overbrace{6a + 6b}^{\text{divisible by 6}} + 3 = 6 \times (a + b) + 3 }
According to the resulting algebraic expression, the sum leaves a remainder of three when divided by six.
b) The product of the two numbers:
\latex{ (6a + 1)(6b + 2) = 6a \times 6b + 6a \times 2 + 1 \times 6b + 1 \times 2 = }
\latex{ =36ab + 12a + 6b+ 2 = 6 \times (6ab + 2a + b) + 2 }
\latex{ =36ab + 12a + 6b+ 2 = 6 \times (6ab + 2a + b) + 2 }
According to this algebraic expression, the product of the two numbers leaves a remainder of two when divided by six.
Information: In most cases, when multiplying algebraic expressions consisting of multiple terms, the number of resulting terms is equal to the product of the numbers of terms in the expressions being multiplied.
\latex{ \underbrace{(x + y)}_{\text{2\ terms}}\underbrace{(a + b)}_{\text{2\ terms}} = \underbrace{xa + xb + ya + yb}_{\text{4 terms}}} \latex{ 2 \times 2 = 4 }
\latex{ \underbrace{(x + y)}_{\text{2 terms}}\underbrace{(a + b + c)}_{\text{3 terms}} = \underbrace{xa + xb + xc + ya + yb + yc}_{\text{6 terms}}} \latex{ 2\times 3 = 6 }
If there are like terms in the resulting expression, combine them.
Example 4
Solve the equation \latex{ (x-1)\times (x+2)=x^{2}+8 } over the set of rational numbers.
Solution
\latex{ (x-1)\times (x+2)=x^{2} +8 }
Perform the multiplication:
\latex{ x\times x+x\times 2+(-1)\times x+(-1)\times 2=x^{2}+8 }
\latex{ x^{2}+2x-x-2=x^{2}+8 }
\latex{ x^{2}+x-2=x^{2}+8 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\ / -x^{2} }
\latex{ x^{2}+x-2=x^{2}+8 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\ / -x^{2} }
\latex{ x -2 = 8 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\ / +2 }
\latex{ x=10 } rational number
Check:
left side: \latex{ (10 - 1) \times (10 + 2) = 9 \times12 = 108 };
right side: \latex{ 10^{2}+8=100+8=108 } .
right side: \latex{ 10^{2}+8=100+8=108 } .
Answer: The solution to the equation is \latex{\underline{\underline{ x = 10}} }.
\latex{ (-1) \times x = -x }
The equation in the example contained the term \latex{ x^2 }, but it was eliminated during simplification.
Exercises
{{exercise_number}}. Write as many different algebraic expressions as you can to represent the area of the following rectangles.
\latex{ x }
a)
b)
c)
\latex{ 5x }
\latex{ 2 }
\latex{ 2b }
\latex{ a }
\latex{ x }
\latex{ 2 }
\latex{ 1 }
\latex{ y }
\latex{ y }
\latex{ b }
\latex{ a }
{{exercise_number}}. Draw rectangles similar to those in the previous exercise, and write as many different algebraic expressions as you can to represent their areas.
{{exercise_number}}. Perform the following multiplications, then combine the like terms where possible.
- \latex{ (x + 1)(x + 2) }
- \latex{ (x - 1)(x + 3)}
- \latex{(2x + 1)(x + 1)}
- \latex{ (2x - 1)(x + 2) }
- \latex{ (a + b)(x + 2a) }
- \latex{ (a - b)(2a + 1) }
- \latex{ (y - 1)(y + 1) }
- \latex{ (a + b)(a + b) }
- \latex{ (3a - 2b)(2a - 3b) }
- \latex{ (2k - 1)(2k + 1)}
- \latex{ (a - 1)(a^2 + 1) }
- \latex{ (x^2y - 1)(y^2 + x) }
- \latex{ (x^2 + y)(y^2 - x)}
- \latex{ (2a^2 + b^2)(2a - b)}
- \latex{(5xa - b^2)(2xa + b)}
{{exercise_number}}. Perform the following multiplications of algebraic expressions formed by multiple terms, then combine the like terms where possible.
- \latex{ (x + 2)(2x + y + 1) }
- \latex{ (a + b)(a + b + 1) }
- \latex{(x + y)(2x + 2y + z) }
- \latex{ (5x - 1)(6x + y + 2) }
- \latex{ (x^2 - 1)(x + y - 2) }
- \latex{ (z - y)(a + y + z) }
{{exercise_number}}. Apricot yoghurt is sold in two types of packaging: one contains \latex{ a } yoghurts, and the other contains \latex{ b } yoghurts. Each \latex{ day, } one store buys \latex{ x } packages of the first type, while another store buys \latex{ y } packages of the second type. How many yoghurts are delivered to the two stores each \latex{ day? }
{{exercise_number}}. A total of \latex{ n } participants took part in an open-water swimming competition, \latex{ k } of whom came from other countries. The entry fee was \latex{ x } \latex{ euros } \latex{ per } \latex{ person, } of which \latex{ y } \latex{ euros } went to the organisers. The remaining amount was used to cover the cost of the ferry that transported the participants back to shore. How much did the local competitors pay in total for the ferry?
{{exercise_number}}. There are \latex{ x } Floreana giant tortoises living in the wild and \latex{ y } in zoos. (One subspecies is represented by only a single individual, known as Lonesome George.) Each tortoise eats an average of \latex{ a } \latex{ kilograms } of grass and \latex{ b } \latex{ kilograms } of leaves \latex{ per } \latex{ year. } How many \latex{ kilograms } of plants do these tortoises consume in one \latex{ year? }
{{exercise_number}}. Match the algebraic expressions in row \latex{ A } with their equivalent expressions in row \latex{ B }.
\latex{ (x+7)(2x-1) }
\latex{ x^2 +x-6}
\latex{ (x-2)(x+3) }
\latex{ (x-2)(x+2) }
\latex{ (2x+2)(x-3) }
\latex{ x(x^2 - 5) }
\latex{ 2x^2 + 13x - 7 }
\latex{ x^3 -5x }
\latex{ 2x^2 - 4x - 6 }
\latex{ x^2 - 4 }
\latex{ A }
\latex{ B }
{{exercise_number}}. Fill in the boxes with the appropriate cards so that each equation is true over the set of rational numbers. Write out the completed equations.
\latex{ \fcolorbox{#feb811}{#fef4dc}{\color{#000000}{$2a^\text{2}$}} }; \latex{ \fcolorbox{#feb811}{#fef4dc}{\color{#000000}{$a^\text{2}$}} }; \latex{ \fcolorbox{#feb811}{#fef4dc}{\textcolor{#000000}{2}}};\latex{ \fcolorbox{#feb811}{#fef4dc}{\textcolor{#000000}{$4a$}}}; \latex{ \fcolorbox{#feb811}{#fef4dc}{\textcolor{#000000}{$a$}}}; \latex{ \fcolorbox{#feb811}{#fef4dc}{\textcolor{#000000}{$13a$}}}; \latex{ \fcolorbox{#feb811}{#fef4dc}{\textcolor{#000000}{$2a$}}}
- \latex{ (a + 1)(a - 1) = \fcolorbox{#006cb7}{#eaf1fa}{\textcolor{#eaf1fa}{O}} -1}
- \latex{(a + 3)(a - 2) = a^2 + \fcolorbox{#006cb7}{#eaf1fa}{\textcolor{#eaf1fa}{O}} -6}
- \latex{(2a + 3)(3a + 2) = 6a^2+ \fcolorbox{#006cb7}{#eaf1fa}{\textcolor{#eaf1fa}{O}} +6}
- \latex{ (2a + 1)(a - 3) = \fcolorbox{#006cb7}{#eaf1fa}{\textcolor{#eaf1fa}{O}}- 5a - 3}
- \latex{ (1 + 3a)(2 - a) = -3a^2 + 5a+ \fcolorbox{#006cb7}{#eaf1fa}{\textcolor{#eaf1fa}{O}}}
- \latex{ (a + 2)(a + 2) = 4 + \fcolorbox{#006cb7}{#eaf1fa}{\textcolor{#eaf1fa}{O}}+a^2}
{{exercise_number}}. Two numbers are divided by seven. The first number leaves a remainder of five, and the second number leaves a remainder of six. What is the remainder when the following are divided by seven:
a) their sum, b) their difference, c) their product?
{{exercise_number}}. Solve the following equations.
- \latex{ (x - 1)(x + 1) = x^2 + 2x + 1 }
- \latex{ (2x - 3)(x - 1) = (x + 2)(2x- 1) }
- \latex{ (x + 2)(x - 3) = (x - 2)(x + 3) + 2x }
- \latex{ (2x + 2)(1 - 2x) = (3 + x)(1 - 4x) }
Quiz
If the notation \latex{ 60/306 } represents \latex{ 29 } February, what date does the notation \latex{ 306/60 } represent?
