Factor f(x)=64x^3-240x^2+300x-125 ?

$f (x) = 64 x^{3} - 240 x^{2} + 300 x - 125$ ?

Factor $64 x^{3} - 240 x^{2} + 300 x - 125$ using the rational roots test.

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 125, \pm 5, \pm 25 q = \pm 1, \pm 64, \pm 2, \pm 32, \pm 4, \pm 16, \pm 8$

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 0.015625, \pm 0.5, \pm 0.03125, \pm 0.25, \pm 0.0625, \pm 0.125, \pm 125, \pm 1.953125, \pm 62.5, \pm 3.90625, \pm 31.25, \pm 7.8125, \pm 15.625, \pm 5, \pm 0.078125, \pm 2.5, \pm 0.15625, \pm 1.25, \pm 0.3125, \pm 0.625, \pm 25, \pm 0.390625, \pm 12.5, \pm 0.78125, \pm 6.25, \pm 1.5625, \pm 3.125$

Substitute $1.25$ and simplify the expression. In this case, the expression is equal to $0$ so $1.25$ is a root of the polynomial.

Substitute $1.25$ into the polynomial.

$64 \cdot {1.25}^{3} - 240 \cdot {1.25}^{2} + 300 \cdot 1.25 - 125$

Raise $1.25$ to the power of $3$ .

$64 \cdot 1.953125 - 240 \cdot {1.25}^{2} + 300 \cdot 1.25 - 125$

Multiply $64$ by $1.953125$ .

$125 - 240 \cdot {1.25}^{2} + 300 \cdot 1.25 - 125$

Raise $1.25$ to the power of $2$ .

$125 - 240 \cdot 1.5625 + 300 \cdot 1.25 - 125$

Multiply $- 240$ by $1.5625$ .

$125 - 375 + 300 \cdot 1.25 - 125$

Subtract $375$ from $125$ .

$- 250 + 300 \cdot 1.25 - 125$

Multiply $300$ by $1.25$ .

$- 250 + 375 - 125$

Add $- 250$ and $375$ .

$125 - 125$

Subtract $125$ from $125$ .

$0$

Since $1.25$ is a known root, divide the polynomial by $4 x - 5$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{64 x^{3} - 240 x^{2} + 300 x - 125}{4 x - 5}$

Divide $64 x^{3} - 240 x^{2} + 300 x - 125$ by $4 x - 5$ .

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$4 x$

$5$

$64 x^{3}$

$240 x^{2}$

$300 x$

$125$

Divide the highest order term in the dividend $64 x^{3}$ by the highest order term in divisor $4 x$ .

					$16 x^{2}$
	$4 x$	-	$5$		$64 x^{3}$	-	$240 x^{2}$	+	$300 x$	-	$125$

Multiply the new quotient term by the divisor.

				$16 x^{2}$
$4 x$	-	$5$		$64 x^{3}$	-	$240 x^{2}$	+	$300 x$	-	$125$
			+	$64 x^{3}$	-	$80 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $64 x^{3} - 80 x^{2}$

				$16 x^{2}$
$4 x$	-	$5$		$64 x^{3}$	-	$240 x^{2}$	+	$300 x$	-	$125$
			-	$64 x^{3}$	+	$80 x^{2}$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$16 x^{2}$
$4 x$	-	$5$		$64 x^{3}$	-	$240 x^{2}$	+	$300 x$	-	$125$
			-	$64 x^{3}$	+	$80 x^{2}$
					-	$160 x^{2}$

Pull the next terms from the original dividend down into the current dividend.

				$16 x^{2}$
$4 x$	-	$5$		$64 x^{3}$	-	$240 x^{2}$	+	$300 x$	-	$125$
			-	$64 x^{3}$	+	$80 x^{2}$
					-	$160 x^{2}$	+	$300 x$

Divide the highest order term in the dividend $- 160 x^{2}$ by the highest order term in divisor $4 x$ .

				$16 x^{2}$	-	$40 x$
$4 x$	-	$5$		$64 x^{3}$	-	$240 x^{2}$	+	$300 x$	-	$125$
			-	$64 x^{3}$	+	$80 x^{2}$
					-	$160 x^{2}$	+	$300 x$

Multiply the new quotient term by the divisor.

				$16 x^{2}$	-	$40 x$
$4 x$	-	$5$		$64 x^{3}$	-	$240 x^{2}$	+	$300 x$	-	$125$
			-	$64 x^{3}$	+	$80 x^{2}$
					-	$160 x^{2}$	+	$300 x$
					-	$160 x^{2}$	+	$200 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 160 x^{2} + 200 x$

				$16 x^{2}$	-	$40 x$
$4 x$	-	$5$		$64 x^{3}$	-	$240 x^{2}$	+	$300 x$	-	$125$
			-	$64 x^{3}$	+	$80 x^{2}$
					-	$160 x^{2}$	+	$300 x$
					+	$160 x^{2}$	-	$200 x$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$16 x^{2}$	-	$40 x$
$4 x$	-	$5$		$64 x^{3}$	-	$240 x^{2}$	+	$300 x$	-	$125$
			-	$64 x^{3}$	+	$80 x^{2}$
					-	$160 x^{2}$	+	$300 x$
					+	$160 x^{2}$	-	$200 x$
							+	$100 x$

Pull the next terms from the original dividend down into the current dividend.

				$16 x^{2}$	-	$40 x$
$4 x$	-	$5$		$64 x^{3}$	-	$240 x^{2}$	+	$300 x$	-	$125$
			-	$64 x^{3}$	+	$80 x^{2}$
					-	$160 x^{2}$	+	$300 x$
					+	$160 x^{2}$	-	$200 x$
							+	$100 x$	-	$125$

Divide the highest order term in the dividend $100 x$ by the highest order term in divisor $4 x$ .

				$16 x^{2}$	-	$40 x$	+	$25$
$4 x$	-	$5$		$64 x^{3}$	-	$240 x^{2}$	+	$300 x$	-	$125$
			-	$64 x^{3}$	+	$80 x^{2}$
					-	$160 x^{2}$	+	$300 x$
					+	$160 x^{2}$	-	$200 x$
							+	$100 x$	-	$125$

Multiply the new quotient term by the divisor.

				$16 x^{2}$	-	$40 x$	+	$25$
$4 x$	-	$5$		$64 x^{3}$	-	$240 x^{2}$	+	$300 x$	-	$125$
			-	$64 x^{3}$	+	$80 x^{2}$
					-	$160 x^{2}$	+	$300 x$
					+	$160 x^{2}$	-	$200 x$
							+	$100 x$	-	$125$
							+	$100 x$	-	$125$

The expression needs to be subtracted from the dividend, so change all the signs in $100 x - 125$

				$16 x^{2}$	-	$40 x$	+	$25$
$4 x$	-	$5$		$64 x^{3}$	-	$240 x^{2}$	+	$300 x$	-	$125$
			-	$64 x^{3}$	+	$80 x^{2}$
					-	$160 x^{2}$	+	$300 x$
					+	$160 x^{2}$	-	$200 x$
							+	$100 x$	-	$125$
							-	$100 x$	+	$125$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$16 x^{2}$	-	$40 x$	+	$25$
$4 x$	-	$5$		$64 x^{3}$	-	$240 x^{2}$	+	$300 x$	-	$125$
			-	$64 x^{3}$	+	$80 x^{2}$
					-	$160 x^{2}$	+	$300 x$
					+	$160 x^{2}$	-	$200 x$
							+	$100 x$	-	$125$
							-	$100 x$	+	$125$
										$0$

Since the remander is $0$ , the final answer is the quotient.

$16 x^{2} - 40 x + 25$

Write $64 x^{3} - 240 x^{2} + 300 x - 125$ as a set of factors.

$f (x) = (4 x - 5) (16 x^{2} - 40 x + 25)$

Factor using the perfect square rule.

Rewrite $16 x^{2}$ as ${(4 x)}^{2}$ .

$f (x) = (4 x - 5) ({(4 x)}^{2} - 40 x + 25)$

Rewrite $25$ as $5^{2}$ .

$f (x) = (4 x - 5) ({(4 x)}^{2} - 40 x + 5^{2})$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$40 x = 2 \cdot (4 x) \cdot 5$

Rewrite the polynomial.

$f (x) = (4 x - 5) ({(4 x)}^{2} - 2 \cdot (4 x) \cdot 5 + 5^{2})$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 4 x$ and $b = 5$ .

$f (x) = (4 x - 5) {(4 x - 5)}^{2}$

Combine like factors.

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (x) = {(4 x - 5)}^{1 + 2}$

Add $1$ and $2$ .

$f (x) = {(4 x - 5)}^{3}$

Factor f(x)=64x^3-240x^2+300x-125 ?

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