Multiply or divide, as indicated. (y3 + y2)/7 * 49/(y4 + y3)

Ch. R - Review of Basic Concepts

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Lial 13th Edition

Ch. R - Review of Basic Concepts

문제 38a

1장, 문제 38a

Multiply or divide, as indicated. (y³ + y²)/7 * 49/(y⁴ + y³)

검증된 단계별 안내

Start by writing the expression clearly: \(\frac{y^3 + y^2}{7} \times \frac{49}{y^4 + y^3}\).

Factor the polynomials in the numerators and denominators where possible. For example, factor \(y^3 + y^2\) as \(y^2(y + 1)\) and \(y^4 + y^3\) as \(y^3(y + 1)\).

Rewrite the expression using the factored forms: \(\frac{y^2(y + 1)}{7} \times \frac{49}{y^3(y + 1)}\).

Multiply the numerators together and the denominators together: \(\frac{y^2(y + 1) \times 49}{7 \times y^3(y + 1)}\).

Simplify the expression by canceling common factors such as \((y + 1)\) and reducing the numerical coefficients (e.g., \(\frac{49}{7}\)), and then simplify the powers of \(y\) using the laws of exponents.

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Polynomial Factorization

Polynomial factorization involves breaking down polynomials into simpler expressions called factors. Recognizing common factors, such as y^3 in y^3 + y^2 or y^3 in y^4 + y^3, helps simplify expressions and is essential before performing multiplication or division.

Multiplication and Division of Rational Expressions

Rational expressions are fractions with polynomials in numerator and denominator. Multiplying or dividing them requires multiplying numerators and denominators respectively, and simplifying by canceling common factors to reduce the expression to simplest form.

Simplifying Algebraic Fractions

Simplifying algebraic fractions means reducing them by canceling common factors in numerator and denominator. This process makes expressions easier to work with and is crucial after multiplication or division to find the simplest equivalent form.