Multiplying Polynomials
Multiplying polynomialsA monomial or sum of monomials, like `4x^2 + 3x - 10`. involves applying the rules of exponents and the distributive property to simplify the product. This multiplication can also be illustrated with an area modelA graphic representation of a multiplication problem, in which the length and width of a rectangle are the factors and the area is the product. and can be useful in modeling real world situations. Understanding polynomial products is an important step in factoring and solving algebraic equations.
The distributive property can be used to multiply a polynomial by a monomialA number, a variable, or a product of a number and one or more variables with whole number exponents, such as `-5`, `x`, and `8xy^3`. . Just remember that the monomial must be multiplied by each term in the polynomial. Consider the expression `2x(2x^2 + 5x + 10)`.
This expression can be modeled with a sketch like the one below. This model is called an area model because the rectangular pieces represent the area created by the multiplication of the monomial and the polynomial.
We can see that the product of the width, `2x`, and the length, `2x^2 + 5x + 10`, is the area of the entire shaded region. The area can be split into three smaller pieces. Each of those pieces has a width of `2x` and a length represented by one of the terms of the polynomial.
Area models are a helpful way to visualize a multiplication problem. But we can also find the product of two polynomials algebraically, by applying the distributive property. Remember that the distributive property says that multiplying a sum by a number is the same as multiplying each addend by the number and then adding: `a(b + c) = ab+ac`. It doesn't matter how many terms there are: `a(b + c + d) = ab+ac + ad`.
Let's try one:
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Example |
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Problem |
`5x^3(4x^2 + 3x + 7)` |
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`(5x^3*4x^2)+(5x^3*3x)+(5x^3*7)` |
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Distribute the monomial to each term of the polynomial. |
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`20x^5+15x^4+35x^3` |
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Add the products. |
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Answer |
`20x^5+15x^4+35x^3` |
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Now let's explore multiplying two binomials. Once again, we can draw an area model to help us make sense of the process. We'll use each binomial as one of the dimensions of a rectangle, and their product as the area.
The model below shows `(x + 4)(2x + 2)`:
Each binomial is expanded into individual variables and numbers, `x + 4` along the top of the model and `2x + 2` along the left side. The product of each pair of terms is a colored rectangle. The total area is the sum of all of these small rectangles, which is also the final product of multiplying the binomials. If we combine all the like terms, we can write the product, or area, as `2x^2 + 10x + 8`.
We can also use algebra to determine the product of two binomials. Just multiply each term in one binomial by all the terms in the other binomial as shown below:
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Example |
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Problem |
`(x + 4)(2x + 2)` |
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`x(2x + 2) + 4(2x+2)`
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Multiply each term in one binomial by each term in the other binomial. |
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`2x^2 + 2x + 8x + 8`
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Rewrite to group like terms together. |
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`2x^2 + 10x + 8`
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Combine like terms. |
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Answer |
`2x^2 + 10x + 8` |
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Look back at the rectangle and see where each piece of `2x^2 + 2x + 8x + 8` comes from. Can you see where we multiply `x` by `2x + 2`, and where we get `2x^2` from `x*2x`?
Because multiplication is commutative, the terms can be multiplied in either order. The expression `(2x + 2)(x + 4)` has the same product as `(x + 4)(2x + 2)`, both having a product of `2x^2 + 10x + 8`. (Work it out and see.) The order in which we multiply binomials does not matter. What matters is that we multiply each term in one binomial by each term in the other binomial.
The last step in multiplying polynomials is to combine like terms. Remember that a polynomial is simplified only when there are no like terms remaining.
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Find the product: `(a + 10)(2a - 7)`
A) `2a^2 + 19a - 70`
B) `3a + 3`
C) `2a^2 - 70`
D) `2a^2 + 13a - 70`
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Multiplication of binomials is sometimes needed to solve geometry problems. Suppose we want to find the area of a triangle with a base of `4x - 10` and a height of `2x + 3`. To find the area of the triangle, we find the product of `1/2` of the base and the height (that's the area formula for triangles). This can be shown by the expression `1/2(4x - 10)(2x + 3)`.
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Example |
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Problem |
`1/2(4x - 10)(2x + 3)` |
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`1/2[4x(2x + 3) - 10(2x+3)]` |
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Distribute multiplication. |
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`1/2[8x^2 + 12x - 20x - 30]`
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Multiply. Be careful about the negative sign when distributing `-10` to `2x+3`. |
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`1/2[8x^2 - 8x - 30]`
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Combine like terms. |
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`8/2x^2-8/2x-30/2` |
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Simplify. |
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Answer |
`4x^2 - 4x-15` |
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Another type of polynomial multiplication problem is the product of a binomial and trinomial. Here again, the process is the same as with the other types of problems. Each term in the binomial must be multiplied by each of the terms in the trinomial. Two examples are shown below.
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Example |
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Problem |
`(3x + 6)(5x^2 + 3x-10)` |
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`3x(5x^2 + 3x - 10) + 6(5x^2+3x - 10)` |
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Multiply each term in the binomial by the polynomial. |
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`(15x^3 + 9x^2 - 30x) + (30x^2 + 18x - 60)` |
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Use distributive property. |
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`15x^3 + 9x^2 - 30x + 30x^2 + 18x - 60` |
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Rewrite without parentheses. |
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`15x^3 + 9x^2 + 30x^2 - 30x + 18x - 60` |
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Regroup like terms to combine. |
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`15x^3 + 39x^2 - 12x - 60` |
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Combine like terms. |
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Answer |
`15x^3 + 39x^2 - 12x - 60` |
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The next example shows multiplication by a binomial and polynomial that each contain subtraction.
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Example |
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Problem |
`(2p - 1)(3p^2 - 3p+1)` |
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`2p(3p^2 - 3p+1)-1(3p^2-3p+1)` |
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Multiply each term in the binomial by the polynomial. |
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`(6p^3-6p^2+2p) - 1(3p^2) - 1(-3p) - 1(1)` |
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Be careful of the negative sign when distributing `-1` to the trinomial. |
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`6p^3-6p^2+2p - 3p^2 + 3p - 1` |
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Rewrite without parentheses. |
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`6p^3 - 6p^2-3p^2 + 2p + 3p - 1` |
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Regroup like terms to combine. |
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`6p^3 - 9p^2+5p - 1` |
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Combine like terms. |
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Answer |
`6p^3 - 9p^2+5p - 1` |
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Find the product: `(3x - 2)(2x^2 + 4x - 11)`
A) `6x^3 + 8x^2 - 41x + 22`
B) `6x^3 + 8x^2 - 41x - 22`
C) `6x^3 + 12x + 22`
D) `3x^3 + 8x^2 + 25x - 22`
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Multiplication of binomials and polynomials requires use of the distributive property and integer operations. Whether the polynomials are monomials, binomials, or trinomials, carefully multiply each term in one polynomial by each term in the other polynomial. Be careful to watch the addition and subtraction signs and negative coefficients. A product is written in simplified form if all of its like terms have been combined.