Sure, let's go through this step-by-step to match each polynomial to its correct name.
1. Polynomial [tex]\( h(x) = 15x + 2 \)[/tex]:
- This polynomial is of the form [tex]\( ax + b \)[/tex].
- It is a linear polynomial because the highest power of [tex]\( x \)[/tex] is 1.
- It has two terms, hence it is a binomial.
- So, [tex]\( h(x) = 15x + 2 \)[/tex] is a Linear binomial.
- Therefore, [tex]\( h(x) \)[/tex] corresponds to option 2.
2. Polynomial [tex]\( f(x) = x^4 - 3x^2 + 9x^2 \)[/tex]:
- First, simplify the polynomial:
[tex]\[
f(x) = x^4 - 3x^2 + 9x^2 = x^4 + 6x^2
\][/tex]
- The highest power of [tex]\( x \)[/tex] is 4, so it is a quartic (4th degree) polynomial.
- It has three terms after simplification, making it a trinomial.
- So, [tex]\( f(x) = x^4 + 6x^2 \)[/tex] is a Quartic trinomial.
- Therefore, [tex]\( f(x) \)[/tex] corresponds to option 3.
3. Polynomial [tex]\( g(x) = -5x^3 \)[/tex]:
- This polynomial has only one term.
- The highest power of [tex]\( x \)[/tex] is 3, making it a cubic (3rd degree) polynomial.
- It is a monomial because it has a single term.
- However, this polynomial is not relevant to the options provided in the question, so we ignore this one for matching.
4. Polynomial [tex]\( f(x) = 3x^2 - 5x + 7 \)[/tex]:
- The highest power of [tex]\( x \)[/tex] is 2, making it a quadratic (2nd degree) polynomial.
- It has three terms, making it a trinomial.
- So, [tex]\( f(x) = 3x^2 - 5x + 7 \)[/tex] is a Quadratic trinomial.
- Therefore, [tex]\( f(x) \)[/tex] corresponds to option 1.
Let's summarize the matches:
- [tex]\( h(x) = 15x + 2 \)[/tex] matches with 2 (Linear binomial)
- [tex]\( f(x) = x^4 + 6x^2 \)[/tex] matches with 3 (Quartic trinomial)
- [tex]\( g(x) = -5x^3 \)[/tex] is irrelevant for this matching.
- [tex]\( f(x) = 3x^2 - 5x + 7 \)[/tex] matches with 1 (Quadratic trinomial)
Thus, the correct matching is:
a: 2
b: 3
d: 1
So the answer is:
[tex]\[ ( 2, 3, 1 ) \][/tex]
