Question

State whether each sentence is true or false. If false, replace the underlined phrase or expression to make a true sentence.

$\mathbf{(}\mathbf{x}\mathbf{+}\mathbf{5}\mathbf{)}\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{2}\mathbf{)}$is the factored form of${\mathit{x}}^{\mathbf{2}}\mathbf{-}\mathbf{3}\mathit{x}\mathbf{-}\mathbf{10}$.

Short answer

The statement is false. $(x-5)(x+2)$is the factored form of$x^2-3x-10$

Step-by-step solution

Step 1. State the concept for multiplication of algebraic expressions.

1. The product of two factors with like signs is positive, and the product of two factors with unlike signs is negative.
2. If x is a variable and m, n are positive integers, then$(x^m\times x^n)=x^{m+n}$

Step 2. State multiplication of Two Binomials.

Let$(a+b)$ and $(c+d)$are two binomials. By using the distributive law of multiplication over addition twice, their product as given below.

$\begin{array}{c}(a+b)(c+d)=(a+b)\times (c+d)\\ =a\times (c+d)+b\times (c+d)\\ =(a\times c+a\times d)+(b\times c+b\times d)\\ =(ac+ad)+(bc+bd)\\ =ac+ad+bc+bd\end{array}$

Step 3. Substitute the values.

Substitute the values.

$\begin{array}{c}(x+5)(x-2)=(x+5)(x+(-2))\\ =x(x-2)+5(x-2)\\ =x^2-2x+5x-10\\ =x^2+3x-10\end{array}$

Hence, the given statement is false.