Question

factor the polynomial expression $x^4 + x^2 - 20$. drag each expression to the correct location on the factored solution. each expression can be used more than once, and not all expressions will be used. $x^4 + x^2 - 20 = (x^2 + \square) (\square + \square) (\square - \square)$ the options are: $x$, $x^2$, $4$, $2$, $5$, $10$, $1$

Explanation:

Step1: Let \( y = x^2 \), rewrite the polynomial.

The polynomial \( x^4 + x^2 - 20 \) can be rewritten as \( y^2 + y - 20 \) by substituting \( y = x^2 \) (since \( x^4=(x^2)^2=y^2 \)).

Step2: Factor the quadratic in \( y \).

We need to factor \( y^2 + y - 20 \). We look for two numbers that multiply to \( -20 \) and add up to \( 1 \). The numbers are \( 5 \) and \( -4 \) because \( 5\times(-4)=-20 \) and \( 5 + (-4)=1 \). So, \( y^2 + y - 20=(y + 5)(y - 4) \).

Step3: Substitute back \( y = x^2 \).

Substituting \( y = x^2 \) back, we get \( (x^2 + 5)(x^2 - 4) \).

Step4: Factor the difference of squares.

The term \( x^2 - 4 \) is a difference of squares, which factors as \( (x + 2)(x - 2) \) (since \( a^2 - b^2=(a + b)(a - b) \), here \( a = x \) and \( b = 2 \)).

Step5: Combine the factors.

Putting it all together, \( x^4 + x^2 - 20=(x^2 + 5)(x + 2)(x - 2) \). So, filling in the blanks: the first blank is \( 5 \), then \( x \), \( 2 \), \( x \), \( 2 \).

Answer:

\( x^4 + x^2 - 20=(x^2 + \boldsymbol{5})(\boldsymbol{x}+ \boldsymbol{2})(\boldsymbol{x}- \boldsymbol{2}) \)