Explore: Polynomial Long Division to Factor (No Remainders) Topic Solutions

Use polynomial long division when the divisor is a linear binomial (or higher degree and greater number of terms than a linear binomial).
Simplify the term with the largest degree out of the expression.
Write the dividend and divisor in standard form.
Polynomial expressions in standard form with the exponents (degree) in descending order.
When the quotient has a remainder of zero, this means that the divisor is a factor of the polynomial (dividend). Or, (divisor)(quotient) = dividend.

Another use for long division is determining polynomial factors when there is no remainder.

Example 3

Divide $7 x^{2} - 38 x - 24$ by x − 6.

Plan
Write the expression with the long division symbol
Simplify from largest to smaller degree terms using the divisor
Write the quotient adhering to place-value by degree

Implement

$x - 6 7 x + 4 7 x^{2} - 38 x - 24 - (7 x^{2} - 42 x) 4 x - 24 - (4 x - 24) 0$

Explain

First, find the value that can be used to eliminate $7 x^{2}$ from the dividend
Place 7x over the linear term in the dividend
Subtract all terms
Next, eliminate 4x
Place + 4 over the constant in the dividend
Subtract all terms

This solution has no remainder. The quotient is (7x + 4) .
This is correct because the product of $(x - 6) (7 x + 4) = 7 x^{2} - 38 x - 24 .$

Example 4

Simplify.
$(6 x^{3} + 19 x^{2} + 7 x - 12) {(3 x^{2} + 5 x - 4)}^{- 1}$

$3 x^{2} + 5 x - 4 2 x + 3 6 x^{3} + 19 x^{2} + 7 x - 12 - (6 x^{3} + 10 x^{2} - 8 x) 9 x^{2} + 15 x - 12 - (9 x^{2} + 15 x - 12) 0$

Note

Explain

Eliminate $6 x^{3}$
Place 2x over the linear term in the dividend
$2 x (3 x^{2} + 5 x - 4) = 6 x^{3} + 10 x^{2} - 8 x$
Subtract all terms
Next eliminate $9 x^{2}$
Place + 3 over the constant in the dividend
Subtract all terms

Example 5

Simplify.

$4 x - 3 5 x - 11 20 x^{2} - 59 x + 33 - (20 x^{2} - 15 x) - 44 x + 33 - (- 44 x + 33) 0$

Check

$(4 x - 3) (5 x - 11) 20 x^{2} - 44 x - 15 x + 33 ✓$

Example 6

Simplify.

$(2 x^{3} - 11 x^{2} + 13 x - \frac{3}{2}) \div (2 x^{2} - 8 x + 1)$

$2 x^{2} - 8 x + 1 x - \frac{3}{2} 2 x^{3} - 11 x^{2} + 13 x - \frac{3}{2} - (2 x^{3} - 8 x^{2} + x) - 3 x^{2} + 12 x - \frac{3}{2} - (\frac{}{} - 3 x^{2} + 12 x - \frac{3}{2}) 0$

Check

$(x - \frac{3}{2}) (2 x^{2} - 8 x + 1) 2 x^{3} - 8 x^{2} + x - 3 x^{2} + 12 x - \frac{3}{2} ✓$