Is it a factor? Check the remainder!
Test whether a binomial is a factor by checking the remainder: if dividing p(x) by (x-a) gives zero remainder, then (x-a) is a confirmed factor via the Factor Theorem.
Key Concepts
Property A polynomial, such as $(x c)$, is considered a factor of another polynomial if the remainder is 0 after division. If the remainder is any non zero value, then it is not a factor.
Is $(x + 2)$ a factor of $(2x^3 x^2 7x + 6)$? Yes, because dividing gives a remainder of $0$. Is $(x + 3)$ a factor of $(6x^3 6x^2 6x + 6)$? No, because the division results in a non zero remainder of $204$.
Think of factors as puzzle pieces that fit perfectly. When you divide one polynomial by another, a remainder of zero means it's a perfect fit—the divisor is a factor! Any other remainder means the pieces don't align correctly, so it's not a factor. This gives you a clear 'yes' or 'no' answer every single time.
Common Questions
What does the Factor Theorem say about polynomial factors?
The Factor Theorem states that (x-a) is a factor of polynomial p(x) if and only if p(a)=0. By the Remainder Theorem, dividing p(x) by (x-a) gives remainder p(a). A zero remainder confirms (x-a) divides evenly.
How do you use synthetic division to check if (x-2) is a factor of x^3-3x^2+4?
Set up synthetic division with divisor 2 and coefficients 1, -3, 0, 4. Carry out the process step by step. A final remainder of 0 confirms (x-2) is a factor. A nonzero remainder means it is not a factor.
What is the next step after confirming a factor?
Once (x-a) is confirmed as a factor, the synthetic division quotient gives the remaining polynomial factor. Factor the quotient further to find all zeros. This step-by-step factoring is how Grade 10 students fully factor higher-degree polynomials.