Question

simplify the following expression. 5h³(5h² − h − 3) 25h? + h⁴ + h

Explanation:

Step1: Distribute \(5h^3\)

Using the distributive property \(a(b + c + d)=ab + ac + ad\), we distribute \(5h^3\) over \(5h^2 - h - 3\):
\[

$$\begin{align*} 5h^3(5h^2 - h - 3)&=5h^3\times5h^2+5h^3\times(-h)+5h^3\times(-3)\\ \end{align*}$$

\]

Step2: Multiply the coefficients and add exponents

For the first term: \(5\times5 = 25\) and \(h^3\times h^2=h^{3 + 2}=h^5\), so \(5h^3\times5h^2 = 25h^5\).
For the second term: \(5\times(- 1)=-5\) and \(h^3\times h=h^{3+1}=h^4\), so \(5h^3\times(-h)=-5h^4\).
For the third term: \(5\times(-3)=-15\) and the variable part is \(h^3\), so \(5h^3\times(-3)=-15h^3\).

So the expanded form is \(25h^5-5h^4 - 15h^3\). Comparing with \(25h^{?}+ \square h^4+\square h^{\square}\), we can see:

• The exponent for \(25h^{\?}\) is \(5\) (since we have \(25h^5\)).
• The coefficient for \(\square h^4\) is \(- 5\) (since we have \(-5h^4\)).
• The coefficient for \(\square h^{\square}\) is \(-15\) and the exponent is \(3\) (since we have \(-15h^3\)).

Answer:

For the exponent of \(25h^{?}\): \(5\)

For the coefficient of \(\square h^4\): \(-5\)

For the coefficient of \(\square h^{\square}\): \(-15\) and the exponent is \(3\)

So filling in the blanks: \(25h^{5}+(-5)h^4+(-15)h^{3}\)