Sure, let's factor the given binomial step by step.
We start with the expression:
[tex]\[ 100 - x^2 \][/tex]
This binomial is a difference of squares. The general form for factoring a difference of squares is:
[tex]\[ a^2 - b^2 = (a - b)(a + b) \][/tex]
In our expression, we can identify \( a^2 \) as 100 and \( b^2 \) as \( x^2 \). Let's find \( a \) and \( b \):
- \( 100 \) is a perfect square which can be written as \( 10^2 \). So, \( a = 10 \).
- \( x^2 \) is already in the form of \( b^2 \), so \( b = x \).
Now we can apply the difference of squares formula:
[tex]\[ 100 - x^2 = (10 - x)(10 + x) \][/tex]
So, the fully factored form of the given binomial \( 100 - x^2 \) is:
[tex]\[ (10 - x)(10 + x) \][/tex]
However, since signs matter in the final expression and the order does not matter in multiplication, the expression can alternatively be written as:
[tex]\[ -(x - 10)(x + 10) \][/tex]
Thus, for the expression \( 100 - x^2 \) our factorization will be:
[tex]\[
(10 - x)(10 + x)
\][/tex]
To directly match the placeholder form in your question:
[tex]\[
100 - x^2 = (10 - x)(10 + x)
\][/tex]
So, the placeholders filled in will be:
[tex]\[
(10 - x)(10 + x)
\][/tex]
