Find all values of b or c that will make the polynomial a perfect square trinomial. 4z²+bz+81

검증된 단계별 안내

Recall that a perfect square trinomial can be written in the form $\left( mz + n \right)^2 = m^2 z^2 + 2mnz + n^2$.

Compare the given polynomial \$4z^2 + bz + 81$ to the general form $m^2 z^2 + 2mnz + n^2$ to identify $m^2$ and $n^2$.

Since the coefficient of $z^2$ is 4, set $m^2 = 4$, which gives $m = 2$ (considering the positive root for simplicity).

Since the constant term is 81, set $n^2 = 81$, which gives $n = 9$ (again, considering the positive root).

Use the middle term formula \$2mnz$ to find $b$: $b = 2 \times m \times n = 2 \times 2 \times 9$, and write the value of $b$ that makes the polynomial a perfect square trinomial.

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주요 개념

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Perfect Square Trinomial

A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial, typically in the form (x + d)^2 = x^2 + 2dx + d^2. Recognizing this form helps identify conditions on coefficients to make the polynomial a perfect square.

Comparing Coefficients

To determine values of variables that make a polynomial a perfect square, compare the given polynomial's coefficients with those of the expanded perfect square form. This method allows solving for unknown coefficients by matching terms.

Factoring Quadratic Expressions

Factoring quadratics involves rewriting the expression as a product of binomials. Understanding how to factor and expand quadratics is essential to verify if a polynomial is a perfect square and to find the necessary coefficient values.

Find all values of b or c that will make the polynomial a perfect square trinomial. 4z2+bz+81