Sure, let's work through this step-by-step.
We are given two polynomials:
[tex]\[ p_1 = 9 a^4 - 7 a^2 + 2 a + 50 \][/tex]
and
[tex]\[ p_2 = 8 a^3 + 2 a^2 - 5 a + 30 \][/tex]
We need to subtract [tex]\( p_2 \)[/tex] from [tex]\( p_1 \)[/tex].
So, we write down the subtraction of the two polynomials:
[tex]\[ (9 a^4 - 7 a^2 + 2 a + 50) - (8 a^3 + 2 a^2 - 5 a + 30) \][/tex]
Next, we distribute the subtraction to each term of the second polynomial:
[tex]\[ 9 a^4 - 7 a^2 + 2 a + 50 - 8 a^3 - 2 a^2 + 5 a - 30 \][/tex]
Now we combine like terms:
1. For the [tex]\( a^4 \)[/tex] term, we only have one term: [tex]\( 9 a^4 \)[/tex].
2. For the [tex]\( a^3 \)[/tex] term, we have: [tex]\( - 8 a^3 \)[/tex].
3. For the [tex]\( a^2 \)[/tex] terms, we combine: [tex]\( -7 a^2 - 2 a^2 = -9 a^2 \)[/tex].
4. For the [tex]\( a \)[/tex] terms, we combine: [tex]\( 2 a + 5 a = 7 a \)[/tex].
5. For the constant terms, we combine: [tex]\( 50 - 30 = 20 \)[/tex].
Thus, the result of the subtraction is:
[tex]\[ 9 a^4 - 8 a^3 - 9 a^2 + 7 a + 20 \][/tex]
So, when you subtract the polynomial [tex]\( 8 a^3 + 2 a^2 - 5 a + 30 \)[/tex] from the polynomial [tex]\( 9 a^4 - 7 a^2 + 2 a + 50 \)[/tex], you get:
[tex]\[ 9 a^4 - 8 a^3 - 9 a^2 + 7 a + 20 \][/tex]
