To identify the correct statement about the polynomial [tex]\( y = 7x^2 + 23x + 6 \)[/tex], we need to determine its zeros (the values of [tex]\( x \)[/tex] that make [tex]\( y = 0 \)[/tex]). From the given information, we know the zeros of the polynomial are [tex]\(-3\)[/tex] and [tex]\(-\frac{2}{7}\)[/tex].
Next, let's examine each statement:
A. The zeros are -3 and [tex]\( -\frac{2}{7} \)[/tex], because [tex]\( y = (x - 3)(7x - 2) \)[/tex].
- If we expand [tex]\( (x - 3)(7x - 2) \)[/tex], it would not give us the original polynomial [tex]\( y = 7x^2 + 23x + 6 \)[/tex]. Therefore, this statement is false.
B. The zeros are 3 and [tex]\( \frac{2}{7} \)[/tex], because [tex]\( y = (x + 3)(7x + 2) \)[/tex].
- If we expand [tex]\( (x + 3)(7x + 2) \)[/tex], it would not give us the original polynomial [tex]\( y = 7x^2 + 23x + 6 \)[/tex]. Moreover, the zeros [tex]\( 3 \)[/tex] and [tex]\( \frac{2}{7} \)[/tex] do not match the given zeros. Hence, this statement is false.
C. The zeros are 3 and [tex]\( \frac{2}{7} \)[/tex], because [tex]\( y = (x - 3)(7x - 2) \)[/tex].
- If we expand [tex]\( (x - 3)(7x - 2) \)[/tex], it would not give us the original polynomial [tex]\( y = 7x^2 + 23x + 6 \)[/tex], and the zeros [tex]\( 3 \)[/tex] and [tex]\( \frac{2}{7} \)[/tex] are incorrect. Therefore, this statement is false.
D. The zeros are -3 and [tex]\( -\frac{2}{7} \)[/tex], because [tex]\( y = (x + 3)(7x + 2) \)[/tex].
- Let's expand [tex]\( (x + 3)(7x + 2) \)[/tex]:
[tex]\[ (x + 3)(7x + 2) = x(7x + 2) + 3(7x + 2) = 7x^2 + 2x + 21x + 6 = 7x^2 + 23x + 6. \][/tex]
This is exactly the original polynomial [tex]\( y = 7x^2 + 23x + 6 \)[/tex]. Thus, the correct factorization is [tex]\( y = (x + 3)(7x + 2) \)[/tex], and the zeros are indeed [tex]\(-3\)[/tex] and [tex]\( -\frac{2}{7} \)[/tex].
Since statement D is both factually and mathematically accurate, the correct answer is:
D. The zeros are -3 and [tex]\( -\frac{2}{7} \)[/tex], because [tex]\( y = (x + 3)(7x + 2) \)[/tex].
