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Guias de estudo > College Algebra

Operations on Polynomials

Learning Objectives

  • Identify the degree, leading coefficient, and leading term of a polynomial
  • Add and subtract polynomial expressions
  • Multiply polynomials
The area of the front of the doghouse described in the introduction was 4x2+12x4{x}^{2}+\frac{1}{2}x ft2. This is an example of a polynomial, which is a sum of or difference of terms, each consisting of a variable raised to a nonnegative integer power. A number multiplied by a variable raised to an exponent, such as 384π384\pi , is known as a coefficient. Coefficients can be positive, negative, or zero, and can be whole numbers, decimals, or fractions. Each product aixi{a}_{i}{x}^{i}, such as 384πw384\pi w, is a term of a polynomial. If a term does not contain a variable, it is called a constant. A polynomial containing only one term, such as 5x45{x}^{4}, is called a monomial. A polynomial containing two terms, such as 2x−92x - 9, is called a binomial. A polynomial containing three terms, such as −3x2+8x−7-3{x}^{2}+8x - 7, is called a trinomial. We can find the degree of a polynomial by identifying the highest power of the variable that occurs in the polynomial. The term with the highest degree is called the leading term because it is usually written first. The coefficient of the leading term is called the leading coefficient. When a polynomial is written so that the powers are descending, we say that it is in standard form. A polynomial reading: a sub n times x to the nth power plus and so on plus a sub 2 times x squared plus a sub one times x plus a subzero is shown. The a in the term a sub n is labeled: leading coefficient. The n in the term x to the nth power is labeled: degree. Finally, the entire term is labeled as: Leading term.

A General Note: Polynomials

A polynomial is an expression that can be written in the form
anxn+⋯+a2x2+a1x+a0{a}_{n}{x}^{n}+\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}
Each real number aiis called a coefficient. The number a0{a}_{0} that is not multiplied by a variable is called a constant. Each product aixi{a}_{i}{x}^{i} is a term of a polynomial. The highest power of the variable that occurs in the polynomial is called the degree of a polynomial. The leading term is the term with the highest power, and its coefficient is called the leading coefficient.

How To: Given a polynomial expression, identify the degree and leading coefficient.

  1. Find the highest power of x to determine the degree.
  2. Identify the term containing the highest power of x to find the leading term.
  3. Identify the coefficient of the leading term.

Example: Identifying the Degree and Leading Coefficient of a Polynomial

For the following polynomials, identify the degree, the leading term, and the leading coefficient.
  1. 3+2x2−4x33+2{x}^{2}-4{x}^{3}
  2. 5t5−2t3+7t5{t}^{5}-2{t}^{3}+7t
  3. 6p−p3−26p-{p}^{3}-2

Answer:

  1. The highest power of x is 3, so the degree is 3. The leading term is the term containing that degree, −4x3-4{x}^{3}. The leading coefficient is the coefficient of that term, −4-4.
  2. The highest power of t is 55, so the degree is 55. The leading term is the term containing that degree, 5t55{t}^{5}. The leading coefficient is the coefficient of that term, 55.
  3. The highest power of p is 33, so the degree is 33. The leading term is the term containing that degree, −p3-{p}^{3}, The leading coefficient is the coefficient of that term, −1-1.

Try It

Identify the degree, leading term, and leading coefficient of the polynomial 4x2−x6+2x−64{x}^{2}-{x}^{6}+2x - 6.

Answer: The degree is 6, the leading term is −x6-{x}^{6}, and the leading coefficient is −1-1.

Example: Subtracting Polynomials

Find the difference.

(7x4−x2+6x+1)−(5x3−2x2+3x+2)\left(7{x}^{4}-{x}^{2}+6x+1\right)-\left(5{x}^{3}-2{x}^{2}+3x+2\right)

Answer: 7x4−5x3+(−x2+2x2)+(6x−3x)+(1−2) Combine like terms.7x4−5x3+x2+3x−1Simplify.\begin{array}{cc}7{x}^{4}-5{x}^{3}+\left(-{x}^{2}+2{x}^{2}\right)+\left(6x - 3x\right)+\left(1 - 2\right)\text{ }\hfill & \text{Combine like terms}.\hfill \\ 7{x}^{4}-5{x}^{3}+{x}^{2}+3x - 1\hfill & \text{Simplify}.\hfill \end{array}

Analysis of the Solution

Note that finding the difference between two polynomials is the same as adding the opposite of the second polynomial to the first.

Try It

Find the difference. (−7x3−7x2+6x−2)−(4x3−6x2−x+7)\left(-7{x}^{3}-7{x}^{2}+6x - 2\right)-\left(4{x}^{3}-6{x}^{2}-x+7\right)

Answer: −11x3−x2+7x−9-11{x}^{3}-{x}^{2}+7x - 9

  Watch this video to see more examples of how to use the distributive property to multiply polynomials. https://youtu.be/bwTmApTV_8o

Using FOIL to Multiply Binomials

A shortcut called FOIL is sometimes used to find the product of two binomials. It is called FOIL because we multiply the first terms, the outer terms, the inner terms, and then the last terms of each binomial. Two quantities in parentheses are being multiplied, the first being: a times x plus b and the second being: c times x plus d. This expression equals ac times x squared plus ad times x plus bc times x plus bd. The terms ax and cx are labeled: First Terms. The terms ax and d are labeled: Outer Terms. The terms b and cx are labeled: Inner Terms. The terms b and d are labeled: Last Terms. The FOIL method arises out of the distributive property. We are simply multiplying each term of the first binomial by each term of the second binomial, and then combining like terms.

How To: Given two binomials, use FOIL to simplify the expression.

  1. Multiply the first terms of each binomial.
  2. Multiply the outer terms of the binomials.
  3. Multiply the inner terms of the binomials.
  4. Multiply the last terms of each binomial.
  5. Add the products.
  6. Combine like terms and simplify.

Example: Using FOIL to Multiply Binomials

Use FOIL to find the product.

Answer: Find the product of the first terms. Find the product of the outer terms. Find the product of the inner terms. Find the product of the last terms.

6x2+6x−54x−54Add the products.6x2+(6x−54x)−54Combine like terms.6x2−48x−54Simplify.\begin{array}{cc}6{x}^{2}+6x - 54x - 54\hfill & \text{Add the products}.\hfill \\ 6{x}^{2}+\left(6x - 54x\right)-54\hfill & \text{Combine like terms}.\hfill \\ 6{x}^{2}-48x - 54\hfill & \text{Simplify}.\hfill \end{array}

Try It

Use FOIL to find the product. (x+7)(3x−5)\left(x+7\right)\left(3x - 5\right)

Answer: 3x2+16x−353{x}^{2}+16x - 35

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