Find each product. See Example 5. 4x² (3x³ + 2x² - 5x +1)

검증된 단계별 안내

Identify the expression to be multiplied: $4x^{2} \times (3x^{3} + 2x^{2} - 5x + 1)$.

Apply the distributive property by multiplying \$4x^{2}$ with each term inside the parentheses separately.

Multiply \$4x^{2}$ by the first term: $3x^{3}$. This gives $4x^{2} \times 3x^{3} = 12x^{5}$.

Multiply \$4x^{2}$ by the second term: $2x^{2}$. This gives $4x^{2} \times 2x^{2} = 8x^{4}$.

Continue by multiplying \$4x^{2}$ by the third term $-5x$ and the fourth term $1$, writing each product before combining all terms.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Polynomial Multiplication

Polynomial multiplication involves distributing each term of one polynomial to every term of the other. This process requires multiplying coefficients and adding exponents of like bases, then combining like terms to simplify the expression.

Exponent Rules

When multiplying terms with the same base, add their exponents (e.g., x^a * x^b = x^(a+b)). This rule is essential for correctly handling powers of variables during polynomial multiplication.

Combining Like Terms

After multiplying, terms with the same variable raised to the same power must be combined by adding or subtracting their coefficients. This step simplifies the polynomial to its standard form.