Question

Factor the perfect-square trinomial in y = (x2 + 2x + 1) − 1− 1.

Answer 1

Answer:

(x+1)²

Step-by-step explanation:

Here, the given expression is,

$y=(x^2+2x+1)-1 -1$

$\implies y=(x^2+2x+1)-2$

Since, a perfect-square trinomial is a trinomial that can be expressed as the square of a binomial,

Now, we can write,

$x^2+2x+1=x^2+x+x+1=x(x+1)+1(x+1)=(x+1)(x+1)=(x+1)^2$

Where, x+1 is a binomial,

Now, in the given expression,

The trinomial $x^2+2x+1$ can expressed as the square of a binomial,

Thus, in the given expression,

$x^2+2x+1$ is the perfect-square trinomial

And, its factor form is (x+1)².

Answer 2

The perfect square trinomial in the given expression is $\boxed{y=(x^{2}+2x+1)^{-1}}$ which is also expressed as $\boxed{\left(\dfrac{1}{x+1}\right)^{2}}$.

Further explanation:

A perfect-square trinomial is a trinomial which can be expressed as the square of a binomial.

A trinomial is an expression which contains three terms.

A binomial is an expression which contains two terms.

The given expression is as follows:

$\boxed{y=(x^{2}+2x+1)^{-1}-1}$

The trinomial in the given expression is as follows:

$\boxed{y=(x^{2}+2x+1)^{-1}}$

Any expression which raised to negative powers is expressed as follows:

$(x)^{-1}=\dfrac{1}{x}$  

Therefore, the trinomial in the given expression is expressed as follows:

$\begin{aligned}y&=(x^{2}+2x+1)^{-1}\\&=\dfrac{1}{(x^{2}+2x+1)^{-1}}\end{aligned}$

The identity of $(a+b)^{2}$ is expressed as follows:

$\boxed{(a+b)^{2}=a^{2}+b^{2}+(2\cdot a\cdot b)}$

Expression the trinomial in the given expression as shown below:

$\begin{aligned}y&=(x^{2}+2x+1)^{-1}\\&=\dfrac{1}{(x^{2}+2x+1)}\\&=\dfrac{1}{x^{2}+1^{2}+(2\cdot x\cdot 1)}\end{aligned}$

Use the identity of $(a+b)^{2}$ to simplify the above expression as shown below:

$\begin{aligned}y&=\dfrac{1}{x^{2}+1^{2}+(2\cdot x\cdot 1)}\\&=\dfrac{1}{(x+1)^{2}}\\&=\left(\dfrac{1}{x+1}\right)^{2}\end{aligned}$

Therefore, the perfect square trinomial in the given expression is $\boxed{y=(x^{2}+2x+1)^{-1}}$ which is also expressed as $\boxed{\left(\dfrac{1}{x+1}\right)^{2}}$.

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Answer details:

Grade: Middle school

Subject: Mathematics

Chapter: Polynomial

Keywords:

Trinomials, binomials, monomials, polynomials, variables, exponents,  real numbers, degree of polynomials, equations, expressions, coefficients, zero polynomial, constants, integers, function, domain, range, codomain, graph, abscissa, coordinates, roots of polynomials, bivariate polynomials.