To factor the quadratic polynomial [tex]\( 9 - 6x + x^2 \)[/tex] as a product of two binomials, we can follow these steps:
1. Identify the standard form of the quadratic polynomial: The polynomial [tex]\( 9 - 6x + x^2 \)[/tex] can be written in standard form as [tex]\( ax^2 + bx + c \)[/tex].
- Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -6 \)[/tex], and [tex]\( c = 9 \)[/tex].
2. Look for a pattern: Notice that the given polynomial might be a perfect square trinomial. A perfect square trinomial has the form [tex]\( (ax + b)^2 \)[/tex].
3. Check for perfect square:
- Observe that [tex]\( x^2 \)[/tex] is a perfect square of [tex]\( x \)[/tex].
- Check if [tex]\( 9 \)[/tex] is a perfect square. Notice that [tex]\( 9 = 3^2 \)[/tex].
- Now consider the middle term, [tex]\( -6x \)[/tex]. For perfect square trinomials of the form [tex]\( (ax + b)^2 \)[/tex], the middle term should be [tex]\( 2 \times x \times 3 = 6x \)[/tex]. In this case, the middle term is [tex]\( -6x \)[/tex], which fits perfectly by factoring [tex]\(-3\)[/tex].
4. Write as a binomial squared:
- Since all conditions are met for a perfect square trinomial, we can write the polynomial as [tex]\( (x - 3)^2 \)[/tex].
5. Verify the factorization:
- Expand [tex]\( (x - 3)^2 \)[/tex]:
[tex]\[
(x - 3)^2 = (x - 3)(x - 3) = x^2 - 3x - 3x + 9 = x^2 - 6x + 9
\][/tex]
- This confirms that the factorization is correct.
Therefore, the polynomial [tex]\( 9 - 6x + x^2 \)[/tex] can be factored as:
[tex]\[
( x - 3 )^2
\][/tex]
