Write as a Set of Linear Factors p(x)=49x^4-210x^3+302x^2-74x+5

$p (x) = 49 x^{4} - 210 x^{3} + 302 x^{2} - 74 x + 5$

Step 1

Factor $49 x^{4} - 210 x^{3} + 302 x^{2} - 74 x + 5$ 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.

$p (x) = p = \pm 1, \pm 5$

$q = \pm 1, \pm 49, \pm 7$

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

$p (x) = \pm 1, \pm 0.02040816, \pm 0.\overline{142857}, \pm 5, \pm 0.10204081, \pm 0.\overline{714285}$

Substitute $0.\overline{142857}$ and simplify the expression. In this case, the expression is equal to $0$ so $0.\overline{142857}$ is a root of the polynomial.

Substitute $0.\overline{142857}$ into the polynomial.

$p (x) = 49 \cdot {0.\overline{142857}}^{4} - 210 \cdot {0.\overline{142857}}^{3} + 302 \cdot {0.\overline{142857}}^{2} - 74 \cdot 0.\overline{142857} + 5$

Raise $0.\overline{142857}$ to the power of $4$ .

$p (x) = 49 \cdot 0.00041649 - 210 \cdot {0.\overline{142857}}^{3} + 302 \cdot {0.\overline{142857}}^{2} - 74 \cdot 0.\overline{142857} + 5$

Multiply $49$ by $0.00041649$ .

$p (x) = 0.02040816 - 210 \cdot {0.\overline{142857}}^{3} + 302 \cdot {0.\overline{142857}}^{2} - 74 \cdot 0.\overline{142857} + 5$

Raise $0.\overline{142857}$ to the power of $3$ .

$p (x) = 0.02040816 - 210 \cdot 0.00291545 + 302 \cdot {0.\overline{142857}}^{2} - 74 \cdot 0.\overline{142857} + 5$

Multiply $- 210$ by $0.00291545$ .

$p (x) = 0.02040816 - 0.61224489 + 302 \cdot {0.\overline{142857}}^{2} - 74 \cdot 0.\overline{142857} + 5$

Subtract $0.61224489$ from $0.02040816$ .

$p (x) = - 0.59183673 + 302 \cdot {0.\overline{142857}}^{2} - 74 \cdot 0.\overline{142857} + 5$

Raise $0.\overline{142857}$ to the power of $2$ .

$p (x) = - 0.59183673 + 302 \cdot 0.02040816 - 74 \cdot 0.\overline{142857} + 5$

Multiply $302$ by $0.02040816$ .

$p (x) = - 0.59183673 + 6.1632653 - 74 \cdot 0.\overline{142857} + 5$

Add $- 0.59183673$ and $6.1632653$ .

$p (x) = 5.57142857 - 74 \cdot 0.\overline{142857} + 5$

Multiply $- 74$ by $0.\overline{142857}$ .

$p (x) = 5.57142857 - 10.\overline{571428} + 5$

Subtract $10.\overline{571428}$ from $5.57142857$ .

$p (x) = - 5 + 5$

Add $- 5$ and $5$ .

$p (x) = 0$

Since $0.\overline{142857}$ is a known root, divide the polynomial by $7 x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$p (x) = \frac{49 x^{4} - 210 x^{3} + 302 x^{2} - 74 x + 5}{7 x - 1}$

Divide $49 x^{4} - 210 x^{3} + 302 x^{2} - 74 x + 5$ by $7 x - 1$ .

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

$p (x) =$

Divide the highest order term in the dividend $49 x^{4}$ by the highest order term in divisor $7 x$ .

$p (x) =$

Multiply the new quotient term by the divisor.

$p (x) =$

The expression needs to be subtracted from the dividend, so change all the signs in $49 x^{4} - 7 x^{3}$

$p (x) =$

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

$p (x) =$

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

$p (x) =$

Divide the highest order term in the dividend $- 203 x^{3}$ by the highest order term in divisor $7 x$ .

$p (x) =$

Multiply the new quotient term by the divisor.

$p (x) =$

The expression needs to be subtracted from the dividend, so change all the signs in $- 203 x^{3} + 29 x^{2}$

$p (x) =$

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

$p (x) =$

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

$p (x) =$

Divide the highest order term in the dividend $273 x^{2}$ by the highest order term in divisor $7 x$ .

$p (x) =$

Multiply the new quotient term by the divisor.

$p (x) =$

The expression needs to be subtracted from the dividend, so change all the signs in $273 x^{2} - 39 x$

$p (x) =$

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

$p (x) =$

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

$p (x) =$

Divide the highest order term in the dividend $- 35 x$ by the highest order term in divisor $7 x$ .

$p (x) =$

Multiply the new quotient term by the divisor.

$p (x) =$

The expression needs to be subtracted from the dividend, so change all the signs in $- 35 x + 5$

$p (x) =$

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

$p (x) =$

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

$p (x) = 7 x^{3} - 29 x^{2} + 39 x - 5$

Write $49 x^{4} - 210 x^{3} + 302 x^{2} - 74 x + 5$ as a set of factors.

$p (x) = (7 x - 1) (7 x^{3} - 29 x^{2} + 39 x - 5)$

Step 2

Factor $7 x^{3} - 29 x^{2} + 39 x - 5$ 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.

$p (x) = p = \pm 1, \pm 5$

$q = \pm 1, \pm 7$

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

$p (x) = \pm 1, \pm 0.\overline{142857}, \pm 5, \pm 0.\overline{714285}$

Substitute $0.\overline{142857}$ and simplify the expression. In this case, the expression is equal to $0$ so $0.\overline{142857}$ is a root of the polynomial.

Substitute $0.\overline{142857}$ into the polynomial.

$p (x) = 7 \cdot {0.\overline{142857}}^{3} - 29 \cdot {0.\overline{142857}}^{2} + 39 \cdot 0.\overline{142857} - 5$

Raise $0.\overline{142857}$ to the power of $3$ .

$p (x) = 7 \cdot 0.00291545 - 29 \cdot {0.\overline{142857}}^{2} + 39 \cdot 0.\overline{142857} - 5$

Multiply $7$ by $0.00291545$ .

$p (x) = 0.02040816 - 29 \cdot {0.\overline{142857}}^{2} + 39 \cdot 0.\overline{142857} - 5$

Raise $0.\overline{142857}$ to the power of $2$ .

$p (x) = 0.02040816 - 29 \cdot 0.02040816 + 39 \cdot 0.\overline{142857} - 5$

Multiply $- 29$ by $0.02040816$ .

$p (x) = 0.02040816 - 0.59183673 + 39 \cdot 0.\overline{142857} - 5$

Subtract $0.59183673$ from $0.02040816$ .

$p (x) = - 0.57142857 + 39 \cdot 0.\overline{142857} - 5$

Multiply $39$ by $0.\overline{142857}$ .

$p (x) = - 0.57142857 + 5.57142857 - 5$

Add $- 0.57142857$ and $5.57142857$ .

$p (x) = 5 - 5$

Subtract $5$ from $5$ .

$p (x) = 0$

Since $0.\overline{142857}$ is a known root, divide the polynomial by $7 x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$p (x) = \frac{7 x^{3} - 29 x^{2} + 39 x - 5}{7 x - 1}$

Divide $7 x^{3} - 29 x^{2} + 39 x - 5$ by $7 x - 1$ .

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

$p (x) =$

Divide the highest order term in the dividend $7 x^{3}$ by the highest order term in divisor $7 x$ .

$p (x) =$

Multiply the new quotient term by the divisor.

$p (x) =$

The expression needs to be subtracted from the dividend, so change all the signs in $7 x^{3} - x^{2}$

$p (x) =$

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

$p (x) =$

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

$p (x) =$

Divide the highest order term in the dividend $- 28 x^{2}$ by the highest order term in divisor $7 x$ .

$p (x) =$

Multiply the new quotient term by the divisor.

$p (x) =$

The expression needs to be subtracted from the dividend, so change all the signs in $- 28 x^{2} + 4 x$

$p (x) =$

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

$p (x) =$

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

$p (x) =$

Divide the highest order term in the dividend $35 x$ by the highest order term in divisor $7 x$ .

$p (x) =$

Multiply the new quotient term by the divisor.

$p (x) =$

The expression needs to be subtracted from the dividend, so change all the signs in $35 x - 5$

$p (x) =$

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

$p (x) =$

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

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

Write $7 x^{3} - 29 x^{2} + 39 x - 5$ as a set of factors.

$p (x) = (7 x - 1) ((7 x - 1) (x^{2} - 4 x + 5))$

Remove unnecessary parentheses.

$p (x) = (7 x - 1) (7 x - 1) (x^{2} - 4 x + 5)$

Step 3

Combine like factors.

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

$p (x) = {(7 x - 1)}^{1 + 1} (x^{2} - 4 x + 5)$

Add $1$ and $1$ .

$p (x) = {(7 x - 1)}^{2} (x^{2} - 4 x + 5)$

Write as a Set of Linear Factors p(x)=49x^4-210x^3+302x^2-74x+5

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