Algebra Examples

$(x + 3) {(x - 4)}^{5}$

Step 1

Simplify the polynomial, then reorder it left to right starting with the highest degree term.

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Use the Binomial Theorem.

$(x + 3) (x^{5} + 5 x^{4} \cdot - 4 + 10 x^{3} {(- 4)}^{2} + 10 x^{2} {(- 4)}^{3} + 5 x {(- 4)}^{4} + {(- 4)}^{5})$

Simplify each term.

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Multiply $- 4$ by $5$ .

$(x + 3) (x^{5} - 20 x^{4} + 10 x^{3} {(- 4)}^{2} + 10 x^{2} {(- 4)}^{3} + 5 x {(- 4)}^{4} + {(- 4)}^{5})$

Raise $- 4$ to the power of $2$ .

$(x + 3) (x^{5} - 20 x^{4} + 10 x^{3} \cdot 16 + 10 x^{2} {(- 4)}^{3} + 5 x {(- 4)}^{4} + {(- 4)}^{5})$

Multiply $16$ by $10$ .

$(x + 3) (x^{5} - 20 x^{4} + 160 x^{3} + 10 x^{2} {(- 4)}^{3} + 5 x {(- 4)}^{4} + {(- 4)}^{5})$

Raise $- 4$ to the power of $3$ .

$(x + 3) (x^{5} - 20 x^{4} + 160 x^{3} + 10 x^{2} \cdot - 64 + 5 x {(- 4)}^{4} + {(- 4)}^{5})$

Multiply $- 64$ by $10$ .

$(x + 3) (x^{5} - 20 x^{4} + 160 x^{3} - 640 x^{2} + 5 x {(- 4)}^{4} + {(- 4)}^{5})$

Raise $- 4$ to the power of $4$ .

$(x + 3) (x^{5} - 20 x^{4} + 160 x^{3} - 640 x^{2} + 5 x \cdot 256 + {(- 4)}^{5})$

Multiply $256$ by $5$ .

$(x + 3) (x^{5} - 20 x^{4} + 160 x^{3} - 640 x^{2} + 1280 x + {(- 4)}^{5})$

Raise $- 4$ to the power of $5$ .

$(x + 3) (x^{5} - 20 x^{4} + 160 x^{3} - 640 x^{2} + 1280 x - 1024)$

Expand $(x + 3) (x^{5} - 20 x^{4} + 160 x^{3} - 640 x^{2} + 1280 x - 1024)$ by multiplying each term in the first expression by each term in the second expression.

$x \cdot x^{5} + x (- 20 x^{4}) + x (160 x^{3}) + x (- 640 x^{2}) + x (1280 x) + x \cdot - 1024 + 3 x^{5} + 3 (- 20 x^{4}) + 3 (160 x^{3}) + 3 (- 640 x^{2}) + 3 (1280 x) + 3 \cdot - 1024$

Simplify terms.

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Simplify each term.

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Multiply $x$ by $x^{5}$ by adding the exponents.

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Multiply $x$ by $x^{5}$ .

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Raise $x$ to the power of $1$ .

$x^{1} x^{5} + x (- 20 x^{4}) + x (160 x^{3}) + x (- 640 x^{2}) + x (1280 x) + x \cdot - 1024 + 3 x^{5} + 3 (- 20 x^{4}) + 3 (160 x^{3}) + 3 (- 640 x^{2}) + 3 (1280 x) + 3 \cdot - 1024$

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

$x^{1 + 5} + x (- 20 x^{4}) + x (160 x^{3}) + x (- 640 x^{2}) + x (1280 x) + x \cdot - 1024 + 3 x^{5} + 3 (- 20 x^{4}) + 3 (160 x^{3}) + 3 (- 640 x^{2}) + 3 (1280 x) + 3 \cdot - 1024$

Add $1$ and $5$ .

$x^{6} + x (- 20 x^{4}) + x (160 x^{3}) + x (- 640 x^{2}) + x (1280 x) + x \cdot - 1024 + 3 x^{5} + 3 (- 20 x^{4}) + 3 (160 x^{3}) + 3 (- 640 x^{2}) + 3 (1280 x) + 3 \cdot - 1024$

Rewrite using the commutative property of multiplication.

$x^{6} - 20 x \cdot x^{4} + x (160 x^{3}) + x (- 640 x^{2}) + x (1280 x) + x \cdot - 1024 + 3 x^{5} + 3 (- 20 x^{4}) + 3 (160 x^{3}) + 3 (- 640 x^{2}) + 3 (1280 x) + 3 \cdot - 1024$

Multiply $x$ by $x^{4}$ by adding the exponents.

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Move $x^{4}$ .

$x^{6} - 20 (x^{4} x) + x (160 x^{3}) + x (- 640 x^{2}) + x (1280 x) + x \cdot - 1024 + 3 x^{5} + 3 (- 20 x^{4}) + 3 (160 x^{3}) + 3 (- 640 x^{2}) + 3 (1280 x) + 3 \cdot - 1024$

Multiply $x^{4}$ by $x$ .

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Raise $x$ to the power of $1$ .

$x^{6} - 20 (x^{4} x^{1}) + x (160 x^{3}) + x (- 640 x^{2}) + x (1280 x) + x \cdot - 1024 + 3 x^{5} + 3 (- 20 x^{4}) + 3 (160 x^{3}) + 3 (- 640 x^{2}) + 3 (1280 x) + 3 \cdot - 1024$

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

$x^{6} - 20 x^{4 + 1} + x (160 x^{3}) + x (- 640 x^{2}) + x (1280 x) + x \cdot - 1024 + 3 x^{5} + 3 (- 20 x^{4}) + 3 (160 x^{3}) + 3 (- 640 x^{2}) + 3 (1280 x) + 3 \cdot - 1024$

Add $4$ and $1$ .

$x^{6} - 20 x^{5} + x (160 x^{3}) + x (- 640 x^{2}) + x (1280 x) + x \cdot - 1024 + 3 x^{5} + 3 (- 20 x^{4}) + 3 (160 x^{3}) + 3 (- 640 x^{2}) + 3 (1280 x) + 3 \cdot - 1024$

Rewrite using the commutative property of multiplication.

$x^{6} - 20 x^{5} + 160 x \cdot x^{3} + x (- 640 x^{2}) + x (1280 x) + x \cdot - 1024 + 3 x^{5} + 3 (- 20 x^{4}) + 3 (160 x^{3}) + 3 (- 640 x^{2}) + 3 (1280 x) + 3 \cdot - 1024$

Multiply $x$ by $x^{3}$ by adding the exponents.

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Move $x^{3}$ .

$x^{6} - 20 x^{5} + 160 (x^{3} x) + x (- 640 x^{2}) + x (1280 x) + x \cdot - 1024 + 3 x^{5} + 3 (- 20 x^{4}) + 3 (160 x^{3}) + 3 (- 640 x^{2}) + 3 (1280 x) + 3 \cdot - 1024$

Multiply $x^{3}$ by $x$ .

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Raise $x$ to the power of $1$ .

$x^{6} - 20 x^{5} + 160 (x^{3} x^{1}) + x (- 640 x^{2}) + x (1280 x) + x \cdot - 1024 + 3 x^{5} + 3 (- 20 x^{4}) + 3 (160 x^{3}) + 3 (- 640 x^{2}) + 3 (1280 x) + 3 \cdot - 1024$

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

$x^{6} - 20 x^{5} + 160 x^{3 + 1} + x (- 640 x^{2}) + x (1280 x) + x \cdot - 1024 + 3 x^{5} + 3 (- 20 x^{4}) + 3 (160 x^{3}) + 3 (- 640 x^{2}) + 3 (1280 x) + 3 \cdot - 1024$

Add $3$ and $1$ .

$x^{6} - 20 x^{5} + 160 x^{4} + x (- 640 x^{2}) + x (1280 x) + x \cdot - 1024 + 3 x^{5} + 3 (- 20 x^{4}) + 3 (160 x^{3}) + 3 (- 640 x^{2}) + 3 (1280 x) + 3 \cdot - 1024$

Rewrite using the commutative property of multiplication.

$x^{6} - 20 x^{5} + 160 x^{4} - 640 x \cdot x^{2} + x (1280 x) + x \cdot - 1024 + 3 x^{5} + 3 (- 20 x^{4}) + 3 (160 x^{3}) + 3 (- 640 x^{2}) + 3 (1280 x) + 3 \cdot - 1024$

Multiply $x$ by $x^{2}$ by adding the exponents.

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Move $x^{2}$ .

$x^{6} - 20 x^{5} + 160 x^{4} - 640 (x^{2} x) + x (1280 x) + x \cdot - 1024 + 3 x^{5} + 3 (- 20 x^{4}) + 3 (160 x^{3}) + 3 (- 640 x^{2}) + 3 (1280 x) + 3 \cdot - 1024$

Multiply $x^{2}$ by $x$ .

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Raise $x$ to the power of $1$ .

$x^{6} - 20 x^{5} + 160 x^{4} - 640 (x^{2} x^{1}) + x (1280 x) + x \cdot - 1024 + 3 x^{5} + 3 (- 20 x^{4}) + 3 (160 x^{3}) + 3 (- 640 x^{2}) + 3 (1280 x) + 3 \cdot - 1024$

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

$x^{6} - 20 x^{5} + 160 x^{4} - 640 x^{2 + 1} + x (1280 x) + x \cdot - 1024 + 3 x^{5} + 3 (- 20 x^{4}) + 3 (160 x^{3}) + 3 (- 640 x^{2}) + 3 (1280 x) + 3 \cdot - 1024$

Add $2$ and $1$ .

$x^{6} - 20 x^{5} + 160 x^{4} - 640 x^{3} + x (1280 x) + x \cdot - 1024 + 3 x^{5} + 3 (- 20 x^{4}) + 3 (160 x^{3}) + 3 (- 640 x^{2}) + 3 (1280 x) + 3 \cdot - 1024$

Rewrite using the commutative property of multiplication.

$x^{6} - 20 x^{5} + 160 x^{4} - 640 x^{3} + 1280 x \cdot x + x \cdot - 1024 + 3 x^{5} + 3 (- 20 x^{4}) + 3 (160 x^{3}) + 3 (- 640 x^{2}) + 3 (1280 x) + 3 \cdot - 1024$

Multiply $x$ by $x$ by adding the exponents.

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Move $x$ .

$x^{6} - 20 x^{5} + 160 x^{4} - 640 x^{3} + 1280 (x \cdot x) + x \cdot - 1024 + 3 x^{5} + 3 (- 20 x^{4}) + 3 (160 x^{3}) + 3 (- 640 x^{2}) + 3 (1280 x) + 3 \cdot - 1024$

Multiply $x$ by $x$ .

$x^{6} - 20 x^{5} + 160 x^{4} - 640 x^{3} + 1280 x^{2} + x \cdot - 1024 + 3 x^{5} + 3 (- 20 x^{4}) + 3 (160 x^{3}) + 3 (- 640 x^{2}) + 3 (1280 x) + 3 \cdot - 1024$

Move $- 1024$ to the left of $x$ .

$x^{6} - 20 x^{5} + 160 x^{4} - 640 x^{3} + 1280 x^{2} - 1024 \cdot x + 3 x^{5} + 3 (- 20 x^{4}) + 3 (160 x^{3}) + 3 (- 640 x^{2}) + 3 (1280 x) + 3 \cdot - 1024$

Multiply $- 20$ by $3$ .

$x^{6} - 20 x^{5} + 160 x^{4} - 640 x^{3} + 1280 x^{2} - 1024 x + 3 x^{5} - 60 x^{4} + 3 (160 x^{3}) + 3 (- 640 x^{2}) + 3 (1280 x) + 3 \cdot - 1024$

Multiply $160$ by $3$ .

$x^{6} - 20 x^{5} + 160 x^{4} - 640 x^{3} + 1280 x^{2} - 1024 x + 3 x^{5} - 60 x^{4} + 480 x^{3} + 3 (- 640 x^{2}) + 3 (1280 x) + 3 \cdot - 1024$

Multiply $- 640$ by $3$ .

$x^{6} - 20 x^{5} + 160 x^{4} - 640 x^{3} + 1280 x^{2} - 1024 x + 3 x^{5} - 60 x^{4} + 480 x^{3} - 1920 x^{2} + 3 (1280 x) + 3 \cdot - 1024$

Multiply $1280$ by $3$ .

$x^{6} - 20 x^{5} + 160 x^{4} - 640 x^{3} + 1280 x^{2} - 1024 x + 3 x^{5} - 60 x^{4} + 480 x^{3} - 1920 x^{2} + 3840 x + 3 \cdot - 1024$

Multiply $3$ by $- 1024$ .

$x^{6} - 20 x^{5} + 160 x^{4} - 640 x^{3} + 1280 x^{2} - 1024 x + 3 x^{5} - 60 x^{4} + 480 x^{3} - 1920 x^{2} + 3840 x - 3072$

Simplify by adding terms.

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Add $- 20 x^{5}$ and $3 x^{5}$ .

$x^{6} - 17 x^{5} + 160 x^{4} - 640 x^{3} + 1280 x^{2} - 1024 x - 60 x^{4} + 480 x^{3} - 1920 x^{2} + 3840 x - 3072$

Subtract $60 x^{4}$ from $160 x^{4}$ .

$x^{6} - 17 x^{5} + 100 x^{4} - 640 x^{3} + 1280 x^{2} - 1024 x + 480 x^{3} - 1920 x^{2} + 3840 x - 3072$

Add $- 640 x^{3}$ and $480 x^{3}$ .

$x^{6} - 17 x^{5} + 100 x^{4} - 160 x^{3} + 1280 x^{2} - 1024 x - 1920 x^{2} + 3840 x - 3072$

Subtract $1920 x^{2}$ from $1280 x^{2}$ .

$x^{6} - 17 x^{5} + 100 x^{4} - 160 x^{3} - 640 x^{2} - 1024 x + 3840 x - 3072$

Add $- 1024 x$ and $3840 x$ .

$x^{6} - 17 x^{5} + 100 x^{4} - 160 x^{3} - 640 x^{2} + 2816 x - 3072$

Step 2

The degree of a polynomial is the highest degree of its terms.

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Identify the exponents on the variables in each term, and add them together to find the degree of each term.

$x^{6} \to 6$

$- 17 x^{5} \to 5$

$100 x^{4} \to 4$

$- 160 x^{3} \to 3$

$- 640 x^{2} \to 2$

$2816 x \to 1$

$- 3072 \to 0$

The largest exponent is the degree of the polynomial.

$6$

Step 3

The leading term in a polynomial is the term with the highest degree.

$x^{6}$

Step 4

The leading coefficient of a polynomial is the coefficient of the leading term.

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The leading term in a polynomial is the term with the highest degree.

$x^{6}$

The leading coefficient in a polynomial is the coefficient of the leading term.

$1$

Step 5

List the results.

Polynomial Degree: $6$

Leading Term: $x^{6}$

Leading Coefficient: $1$