To multiply two polynomials [tex]\(f(x) = x^2 + x - 1\)[/tex] and [tex]\(g(x) = x^2 - x + 1\)[/tex], we use the distributive property to expand the product. This means we multiply each term in [tex]\(f(x)\)[/tex] by each term in [tex]\(g(x)\)[/tex] and then combine like terms.
Given:
[tex]\[ f(x) = x^2 + x - 1 \][/tex]
[tex]\[ g(x) = x^2 - x + 1 \][/tex]
We need to find [tex]\( f(x) \times g(x) \)[/tex].
Step-by-step, we distribute each term of [tex]\( f(x) \)[/tex] across every term of [tex]\( g(x) \)[/tex]:
1. Multiply [tex]\( x^2 \)[/tex] by each term in [tex]\( g(x) \)[/tex]:
[tex]\[ x^2 \cdot g(x) = x^2(x^2 - x + 1) = x^4 - x^3 + x^2 \][/tex]
2. Multiply [tex]\( x \)[/tex] by each term in [tex]\( g(x) \)[/tex]:
[tex]\[ x \cdot g(x) = x(x^2 - x + 1) = x^3 - x^2 + x \][/tex]
3. Multiply [tex]\( -1 \)[/tex] by each term in [tex]\( g(x) \)[/tex]:
[tex]\[ -1 \cdot g(x) = -1(x^2 - x + 1) = -x^2 + x - 1 \][/tex]
Now, sum all the resulting terms together:
[tex]\[ (x^4 - x^3 + x^2) + (x^3 - x^2 + x) + (-x^2 + x - 1) \][/tex]
Combine like terms:
[tex]\[ x^4 + (-x^3 + x^3) + (x^2 - x^2 - x^2) + (x + x + x) - 1 \][/tex]
[tex]\[ x^4 + 0 \cdot x^3 - x^2 + 2x - 1 \][/tex]
So, the product of the polynomials [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] is:
[tex]\[ \boxed{x^4 - x^2 + 2x - 1} \][/tex]
This is the simplified polynomial obtained by multiplying [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex].
