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Multiplying Polynomials

Distribute monomials, binomials, and trinomials

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Multiplying Polynomials

The length of a rectangular garden plot is \begin{align*}x^3 + 5x^2 - 1\end{align*}x3+5x21. The width of the plot is \begin{align*}x^2 + 3\end{align*}x2+3. What is the area of the garden plot?

Multiplying Polynomials

Multiplying together polynomials is very similar to multiplying together factors. You can FOIL or we will also present an alternative method. When multiplying together polynomials, you will need to use the properties of exponents, primarily the Product Property \begin{align*}(a^m \cdot a^n = a^{m+n})\end{align*}(aman=am+n) and combine like terms.

Let's find the following products.

  1. \begin{align*}(x^2-5)(x^3 + 2x-9)\end{align*}(x25)(x3+2x9)

Using the FOIL method, you need be careful. First, take the \begin{align*}x^2\end{align*}x2 in the first polynomial and multiply it by every term in the second polynomial.

Now, multiply the -5 and multiply it by every term in the second polynomial.

Lastly, combine any like terms. In this example, only the \begin{align*}x^3\end{align*}x3 terms can be combined.

  1. \begin{align*}(x^2+4x-7)(x^3-8x^2+6x-11)\end{align*}(x2+4x7)(x38x2+6x11)

In this problem, we will use the “box” method. Align the two polynomials along the top and left side of a rectangle and make a row or column for each term. Write the polynomial with more terms along the top of the rectangle.

Multiply each term together and fill in the corresponding spot.

Finally, combine like terms. The final answer is \begin{align*}x^5 -4x^4 -33x^3 + 69x^2 -86x + 77\end{align*}x54x433x3+69x286x+77. This method presents an alternative way to organize the terms. Use whichever method you are more comfortable with. Keep in mind, no matter which method you use, you will multiply every term in the first polynomial by every term in the second.

  1. \begin{align*}(x-5)(2x + 3)(x^2 + 4)\end{align*}(x5)(2x+3)(x2+4)

In this problem we have three binomials. When multiplying three polynomials, start by multiplying the first two binomials together.

\begin{align*}(x-5)(2x+3) &= 2x^2+3x-10x-15\\ &= {\color{red}2x^2-7x-15}\end{align*}(x5)(2x+3)=2x2+3x10x15=2x27x15

Now, multiply the answer by the last binomial.

\begin{align*}({\color{red}2x^2-7x-15})(x^2+4) &= 2x^4+8x^2-7x^3-28x-15x^2-60\\ &= 2x^4-7x^3-7x^2-28x-60\end{align*}(2x27x15)(x2+4)=2x4+8x27x328x15x260=2x47x37x228x60


Example 1

Earlier, you were asked to find the area of the garden plot. 

 Recall that the area of a rectangle is \begin{align*}A = lw\end{align*}, where l is the length and w is the width. Therefore, we need to multiply.

\begin{align*} A =(x^3 + 5x^2 - 1)(x^2 + 3)\\ = x^5 + 3x^3 + 5x^4 + 15x^2 - x^2 - 3\end{align*}

Now combine like terms and simplify. Be sure to write your answer in standard form

\begin{align*}x^5 + 3x^3 + 5x^4 + (15x^2 - x^2) - 3\\ = x^5 + 3x^3 + 5x^4 + 14x^2 - 3\\ = x^5 + 5x^4 + 3x^3 + 14x^2 - 3\end{align*}

Therefore, the area of the garden plot is \begin{align*}x^5 + 5x^4 + 3x^3 + 14x^2 - 3\end{align*}.

Example 2

Find the product: \begin{align*}-2x^2(3x^3-4x^2+12x-9)\end{align*}.

 Use the distributive property to multiply \begin{align*}-2x^2\end{align*} by the polynomial.

\begin{align*}-2x^2(3x^3-4x^2+12x-9) = -6x^5+8x^4-24x^3+18x^2\end{align*}

Example 3

Find the product: \begin{align*}(4x^2-6x+11)(-3x^3+x^2+8x-10)\end{align*}.

 Multiply each term in the first polynomial by each one in the second polynomial.

\begin{align*}(4x^2-6x+11)(-3x^3+x^2+8x-10) &= -12x^5+4x^4+32x^3-40x^2\\ & \qquad \qquad \ +18x^4-6x^3-48x^2+60x\\ & \qquad \qquad \qquad \quad \ -33x^3+11x^2+88x-110\\ &= -12x^5+22x^4-7x^3-77x^2+148x-110\end{align*}

Example 4

Find the product: \begin{align*}(x^2-1)(3x-4)(3x+4)\end{align*}.

 Multiply the first two binomials together.

\begin{align*}(x^2-1)(3x-4) = 3x^3-4x^2-3x+4\end{align*}

Multiply this product by the last binomial.

\begin{align*}(3x^3-4x^2-3x+4)(3x+4) &= 9x^4+12x^3-12x^3-16x^2-9x^2-12x+12x-16\\ &= 9x^4-25x^2-16\end{align*}

Example 5

Find the product: \begin{align*}(2x-7)^2\end{align*}.

The square indicates that there are two binomials. Expand this and multiply.

\begin{align*}(2x-7)^2 &= (2x-7)(2x-7)\\ &= 4x^2-14x-14x+49\\ &= 4x^2-28x+49\end{align*}


Find the product.

  1. \begin{align*}5x(x^2-6x+8)\end{align*}
  2. \begin{align*}-x^2(8x^3-11x+20)\end{align*}
  3. \begin{align*}7x^3(3x^3-x^2+16x+10)\end{align*}
  4. \begin{align*}(x^2+4)(x-5)\end{align*}
  5. \begin{align*}(3x^2-4)(2x-7)\end{align*}
  6. \begin{align*}(9-x^2)(x+2)\end{align*}
  7. \begin{align*}(x^2+1)(x^2-2x-1)\end{align*}
  8. \begin{align*}(5x-1)(x^3+8x-12)\end{align*}
  9. \begin{align*}(x^2-6x-7)(3x^2-7x+15)\end{align*}
  10. \begin{align*}(x-1)(2x-5)(x+8)\end{align*}
  11. \begin{align*}(2x^2+5)(x^2-2)(x+4)\end{align*}
  12. \begin{align*}(5x-12)^2\end{align*}
  13. \begin{align*}-x^4(2x+11)(3x^2-1)\end{align*}
  14. \begin{align*}(4x+9)^2\end{align*}
  15. \begin{align*}(4x^3-x^2-3)(2x^2-x+6)\end{align*}
  16. \begin{align*}(2x^3-6x^2+x+7)(5x^2+2x-4)\end{align*}
  17. \begin{align*}(x^3+x^2-4x+15)(x^2-5x-6)\end{align*}

Answers for Review Problems

To see the Review answers, open this PDF file and look for section 6.5. 

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distributive property

The distributive property states that the product of an expression and a sum is equal to the sum of the products of the expression and each term in the sum. For example, a(b + c) = ab + ac.

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