Learning objectives
Polynomials can be expressed in two primary ways: factored form and standard form. Each form exposes different properties of the function and can be converted from one to the other.
Factored form of a polynomial, which is the product of linear terms, is expressed as f\left(x\right) = a\left(x - r_1\right)\left(x - r_2\right)...\left(x - r_n\right), where r_1, r_2, ..., r_n are the zeros of the function. A benefit of factored form is that the real zeros, found by setting each factor equal to zero and solving for x, correspond to the x-intercepts on the graph of the function. Converting a polynomial to factored form allows us to easily identify these real zeros. To convert a polynomial from factored form to standard form, we multiply out the factors and combine like terms. This is an application of the distributive property.
The standard form of a polynomial is a_n x^n + a_{n - 1} x^{n - 1} + \ldots + a_1 x + a_0, where n is a non-negative integer and each a_i is a coefficient.
This form reveals the end behavior of the function, determined by the sign and degree of the leading term a_nx^n. It also reveals the y-intercept which is a_0.
To multiply polynomials, we apply the distributive property which will require us to use the product of powers property of exponents when multiplying variables:
General example | \left(a_{n}x^{n}+\ldots+a_{0}\right)\left(b_{m}x^{m}+\ldots +b_{0}\right)=\\ \left(a_{n}b_{m}\right)x^{n+m}+\ldots+\left(a_{n}b_0\right)x^n+\ldots+\left(a_{0}b_m\right)x^m+\ldots+\left(a_0b_0\right) |
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Numerical example | \left(5x^2+2x+1\right)\left(x^2-3x+5\right)= 5x^4-13x^3+20x^2+7x+5 |
Because n and m were non-negative integers, n+m will also be non-negative. The exponents will still be constants, so the result is another polynomial.
Write a polynomial expression in standard form for the perimeter of the rectangle shown.
Consider the polynomial function h\left(a\right)=\left(3 + 2a\right)^3 .
Rewrite h\left(a\right) in standard form.
Identify the y-intercept of h\left(a\right).
Describe the end behavior of h\left(x\right).
We can use the distributive property to convert a polynomial in factored form to standard form.
Factoring a polynomial is a process of expressing a polynomial as a product of its factors. In other words, it is the inverse process of multiplying polynomials.
We have learned several different strategies, including a few identities, that can help us factor a polynomial.
Factoring using Greatest Common Factor (GCF):
A greatest common factor from each term of a polynomial is factored out ax+ay+\ldots=a(x+y+\ldots)
Factoring by grouping:
A method for factoring an expression containing at least four terms, by grouping the terms in pairs and taking out common factors ax + ay + bx + by = a\left(x + y\right) + b\left(x + y\right) = \left(x+y\right)\left(a+b\right)
Factoring quadratic trinomials:
A trinomial that can be expressed as the product of two binomials ax^{2} + bx + c= \left(mx + p\right)\left(nx + q\right) where mn=a,pq=c and np+mq=b
Perfect square trinomials:
A trinomial that is formed by multiplying a binomial by itself a^{2} + 2 a b + b^{2} = \left(a + b\right)^{2} \text{ or } a^{2} - 2 a b + b^{2} = \left(a - b\right)^{2}
Difference of two squares:
The result of a perfect square being subtracted from another perfect square a^{2} - b^{2} = \left(a+b\right)\left(a-b\right)
Sum of two cubes:
Two perfect cube expressions being added a^{3} + b^{3} = \left(a+b\right)\left(a^2-ab+b^2\right)
Difference of two cubes:
The result of a perfect cube being subtracted from another perfect cube a^{3} - b^{3} = \left(a-b\right)\left(a^2+ab+b^2\right)
Considerg\left(x\right)=6 x^{4} - 10x^{3} - 24x^{2} in factored form.
Rewrite g\left(x\right) in factored form.
Identify the zeros of g\left(x\right) and their multiplicities.
The factoring methods and identities we can use to fully factor polynomials are
Long division:
In general, when dividing polynomials where the divisor is not a monomial, we can use the process of polynomial long division.
When we divide a dividend, p\left(x\right), by a divisor, b\left(x\right), we can write the expression as the sum of the quotient, q\left(x\right), and the remainder, r\left(x\right), divided by the divisor. The notation for this is {\dfrac{p\left(x\right)}{b\left(x\right)}=q\left(x\right)+\dfrac{r\left(x\right)}{b\left(x\right)}}
If our division was done correctly, then p\left(x\right) = q\left(x\right)b\left(x\right) + r\left(x\right). Polynomial long division works in a very similar way to long division with whole numbers where we:
Note: Before performing long division, the terms of the divisor and dividend should first be arranged in descending order of exponents. In cases where there is no term corresponding to an exponent in the dividend or divisor, we use a placeholder term with a coefficient of 0.
For example, the long division of (2x^3 - 5x + 7)\div(x - 1) is shown below.
For the example above, the dividend is p\left(x\right)=2x^3-5x+7, the divisor is b\left(x\right) =x-1, and we found the quotient to be q(x) =2x^2+2x-3 and remainder r(x) = 4. So, our final answer is \dfrac{2x^3-5x+7}{x-1}=2x^2+2x-3+\dfrac{4}{x-1}
Synthetic division:
There is an efficient method of polynomial division known as synthetic division that can be used only when the divisor is a linear expression in the form x\pm a, where a is a constant.
Consider the long division and synthetic division of \dfrac{3x^3+17x^2+6x-26}{x+5}:
Synthetic division is a short-hand notation version of long division where only the coefficients are used. To begin the synthetic division, we write the coefficients of the terms in the divinded inside an upside down division table. To find the number that goes outside the table, we set the linear divisor equal to zero and solve for x.\begin{aligned}x+5&=0\\x&=-5\end{aligned}
Once the synthetic division is set up, we always bring the first number down. Then, we follow these steps:
The answers beneath the table will be the coefficients of the quotient, and the last number is always the remainder. Because we are dividing by a linear term, the degree of the quotient will be one less than the degree of the divisor.
Therefore, the quotient for the example is q\left(x\right)=3x^2+2x+4 and the remainder is r\left(x\right)=-6, just like we found when we solved using long division.
Write \dfrac{x^3 + 7 x^2 + 14 x + 3}{x + 2} in terms of the quotient and remainder by using long division.
Rewrite \dfrac{2x^3 - 3x^2 + 4x - 1}{x + 1}as the sum of the quotient and a remainder fraction by using synthetic division.
Determine an appropriate method for dividing the following expressions. Explain your choice.
\dfrac{3x^4-6x^3+19x^2-8x+20}{3x^2+4}
\left(-x^4+16x^2+2x\right)\div \left(x-4\right)
\dfrac{4x^2-24x+36}{2x-6}
\dfrac{4x^2+6x-8}{2x+3}
We can divide polynomials using long or synthetic division.
For long division, we first check for any missing terms in the dividend and divisor. If there is a missing term, we add a 0 coefficient in its place. Then, we follow these steps for the division:
Synthetic division is a short-hand notation of division that can only be used when the divisor is a linear expression in the form x\pm a where a is a constant.
We set up the synthetic divison by writing the coefficients of the divident in an upside down division table, and adding 0 coefficients for any missing terms. Then, we set the divisor equal to zero, solve for x, and write the result outside the table. Next, we follow these steps until the last column has been added:
Divide the following polynomials:
Evaluate the following:
Explain the relationship between the remainder of the division and the value of the function at the point.
In previous sections, when dividing p\left(x\right) by a linear divisor, x\pm a where a is a constant, we wrote our answers in the form \dfrac{p\left(x\right)}{x-a}=q\left(x\right)+\dfrac{r\left(x\right)}{x-a}
To check the answers we found for the quotient and remainder, we multiplied the quotient by the divisor and added the remainder. That is, p\left(x\right)=q\left(x\right)\left(x-a\right)+r\left(x\right)
Suppose x=a and the remainder is a constant R. If we substitute this into the above equation, we find \begin{aligned}p\left(a\right)&=q\left(a\right)\left(a-a\right)+R\\&=q\left(a\right)\cdot 0+R\\&=R\end{aligned}
This result shows that we can find the remainder of p\left(x\right)\div \left(x-a\right) without performing the division. It is known as the remainder theorem.
The remainder theorem can be extended to the factor theorem which can help us find factors of a polynomial without having to fully factor the expression.
Using the remainder theorem, determine the remainder when p\left(x\right)=2x^3-4x^2+3x-1 is divided by 2x+1.
When the polynomials P\left(x\right)=x^4+5x^3-mx+n and Q\left(x\right)=mx^2+nx-1 are each divided by D\left(x\right)=x-1, the remainders are 7 and -6 respectively. Find the values of m and n.
Use the Factor theorem to determine if the divisor is a factor of the dividend in the expression: \dfrac{2x^3+x^{2}-10x}{x-2}
Given \left(x+4\right) is a factor of P\left(x\right)=6x^3+31x^2+25x-12, find the remaining factors and rewrite P\left(x\right) as a product of linear factors.
To find the remainder of \dfrac{p\left(x\right)}{x-a} without performing the division, we can evaluate p\left(a\right). The result will be the value of the remainder.
We can use the Factor theorem to determine if a linear factor \left(x-a\right) is a factor of a polynomial p\left(x\right) by determining if p\left(a\right)=0.