5B - Identifying factors of a polynomial

The factor theorem

Consider the factorised polynomial function P(x) = x(x - a)(x - b). Immediately we can see that P(x) = 0 when x = 0, x = a and x = b. From this we can observe the general case which is formalised into the factor theorem:

Testing for simple linear factors

When using the factor theorem it is efficient to test for simple linear factors (x - a) which can be found by testing if P(a) = 0.

• Note: when determining which values to test for, identify factors of the constant term in the polynomial.

5B - VIDEO EXAMPLE 1:

Identify the linear factors of the following polynomial:

VIDEO HERE

5B - VIDEO EXAMPLE 2:

When P(x) = x³ + ax² + x + b is divided by (x + 1) or (x - 2), the remainder is 0. Find the values of a and b.

VIDEO HERE

5B - VIDEO EXAMPLE 3:

Find a and b, given that (x - 3) and (x + 2) are factors of P(x) = x³ + 6x² + ax + b.

VIDEO HERE

The remainder theorem

The remainder theorem can be used to find the remainder of a function if it is divided by some linear factor (polynomials can be divided using long division or synthetic division). The remainder theorem states:

• Note: if the remainder is zero, then the divisor is a factor of the polynomial.

5B - VIDEO EXAMPLE 4:

When P(x) = x³ + 3x² + 7x - b is divided by (x - 2), the remainder is 20. Find the value of b.

VIDEO HERE

5B - VIDEO EXAMPLE 5:

If P(x) = x³ + ax² + bx + 3, P(-1) = 1 and P(1) = 11, find the values of a and b.

VIDEO HERE

The rational-root theorem

When you cannot find a simple factor using the factor theorem, you can use the rational root theorem to find a rational factor. According to the factor theorem:

The rational-root theorem state that if bx + a is factor of P(x) (where the coefficients of the polynomial are integers) then b must divide the leading terms coefficient (the coefficient of the term involving the highest power of x), and a must divide the constant term. Using this theorem we can conduct educated tests to find the rational linear factors.

5B - VIDEO EXAMPLE 6:

Consider the polynomial P(x) = 2x³ - 5x² - x + 6. Use the rational-root theorem to find a rational factor of P(x).

VIDEO HERE