Algebra Examples Simplifying Polynomials

Multiply

$(3 x^{2} + 3 x + 4) (2 x - 1)$

Step 1

Expand $(3 x^{2} + 3 x + 4) (2 x - 1)$ by multiplying each term in the first expression by each term in the second expression.

$3 x^{2} (2 x) + 3 x^{2} \cdot - 1 + 3 x (2 x) + 3 x \cdot - 1 + 4 (2 x) + 4 \cdot - 1$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite using the commutative property of multiplication.

$3 \cdot 2 x^{2} x + 3 x^{2} \cdot - 1 + 3 x (2 x) + 3 x \cdot - 1 + 4 (2 x) + 4 \cdot - 1$

Step 2.1.2

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

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Step 2.1.2.1

Move $x$ .

$3 \cdot 2 (x \cdot x^{2}) + 3 x^{2} \cdot - 1 + 3 x (2 x) + 3 x \cdot - 1 + 4 (2 x) + 4 \cdot - 1$

Step 2.1.2.2

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

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Step 2.1.2.2.1

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

$3 \cdot 2 (x^{1} x^{2}) + 3 x^{2} \cdot - 1 + 3 x (2 x) + 3 x \cdot - 1 + 4 (2 x) + 4 \cdot - 1$

Step 2.1.2.2.2

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

$3 \cdot 2 x^{1 + 2} + 3 x^{2} \cdot - 1 + 3 x (2 x) + 3 x \cdot - 1 + 4 (2 x) + 4 \cdot - 1$

Step 2.1.2.3

Add $1$ and $2$ .

$3 \cdot 2 x^{3} + 3 x^{2} \cdot - 1 + 3 x (2 x) + 3 x \cdot - 1 + 4 (2 x) + 4 \cdot - 1$

Step 2.1.3

Multiply $3$ by $2$ .

$6 x^{3} + 3 x^{2} \cdot - 1 + 3 x (2 x) + 3 x \cdot - 1 + 4 (2 x) + 4 \cdot - 1$

Step 2.1.4

Multiply $- 1$ by $3$ .

$6 x^{3} - 3 x^{2} + 3 x (2 x) + 3 x \cdot - 1 + 4 (2 x) + 4 \cdot - 1$

Step 2.1.5

Rewrite using the commutative property of multiplication.

$6 x^{3} - 3 x^{2} + 3 \cdot 2 x \cdot x + 3 x \cdot - 1 + 4 (2 x) + 4 \cdot - 1$

Step 2.1.6

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

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Step 2.1.6.1

Move $x$ .

$6 x^{3} - 3 x^{2} + 3 \cdot 2 (x \cdot x) + 3 x \cdot - 1 + 4 (2 x) + 4 \cdot - 1$

Step 2.1.6.2

Multiply $x$ by $x$ .

$6 x^{3} - 3 x^{2} + 3 \cdot 2 x^{2} + 3 x \cdot - 1 + 4 (2 x) + 4 \cdot - 1$

Step 2.1.7

Multiply $3$ by $2$ .

$6 x^{3} - 3 x^{2} + 6 x^{2} + 3 x \cdot - 1 + 4 (2 x) + 4 \cdot - 1$

Step 2.1.8

Multiply $- 1$ by $3$ .

$6 x^{3} - 3 x^{2} + 6 x^{2} - 3 x + 4 (2 x) + 4 \cdot - 1$

Step 2.1.9

Multiply $2$ by $4$ .

$6 x^{3} - 3 x^{2} + 6 x^{2} - 3 x + 8 x + 4 \cdot - 1$

Step 2.1.10

Multiply $4$ by $- 1$ .

$6 x^{3} - 3 x^{2} + 6 x^{2} - 3 x + 8 x - 4$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Add $- 3 x^{2}$ and $6 x^{2}$ .

$6 x^{3} + 3 x^{2} - 3 x + 8 x - 4$

Step 2.2.2

Add $- 3 x$ and $8 x$ .

$6 x^{3} + 3 x^{2} + 5 x - 4$

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