Answer :
Let's solve for the value of [tex]\( P \)[/tex] given the equation [tex]\( x^2 + Px - 30 = (x - 15)(x + 2) \)[/tex].
1. Expand the right-hand side:
Begin by expanding the product on the right-hand side.
[tex]\[ (x - 15)(x + 2) \][/tex]
2. Apply the distributive property (also known as the FOIL method for binomials):
[tex]\[ (x - 15)(x + 2) = x \cdot x + x \cdot 2 - 15 \cdot x - 15 \cdot 2 \][/tex]
This simplifies to:
[tex]\[ x^2 + 2x - 15x - 30 \][/tex]
3. Combine like terms:
[tex]\[ x^2 + 2x - 15x - 30 = x^2 - 13x - 30 \][/tex]
4. Compare the expanded form:
Now we have:
[tex]\[ x^2 - 13x - 30 \][/tex]
which needs to be compared to the given polynomial [tex]\( x^2 + Px - 30 \)[/tex].
5. Identify the coefficient of [tex]\( x \)[/tex]:
We can see that in the expression [tex]\( x^2 + Px - 30 \)[/tex], [tex]\( P \)[/tex] is the coefficient of [tex]\( x \)[/tex]. By comparing it to [tex]\( x^2 - 13x - 30 \)[/tex], we can see that the coefficient of [tex]\( x \)[/tex] is [tex]\(-13\)[/tex].
So, the value of [tex]\( P \)[/tex] is:
[tex]\[ P = -13 \][/tex]
1. Expand the right-hand side:
Begin by expanding the product on the right-hand side.
[tex]\[ (x - 15)(x + 2) \][/tex]
2. Apply the distributive property (also known as the FOIL method for binomials):
[tex]\[ (x - 15)(x + 2) = x \cdot x + x \cdot 2 - 15 \cdot x - 15 \cdot 2 \][/tex]
This simplifies to:
[tex]\[ x^2 + 2x - 15x - 30 \][/tex]
3. Combine like terms:
[tex]\[ x^2 + 2x - 15x - 30 = x^2 - 13x - 30 \][/tex]
4. Compare the expanded form:
Now we have:
[tex]\[ x^2 - 13x - 30 \][/tex]
which needs to be compared to the given polynomial [tex]\( x^2 + Px - 30 \)[/tex].
5. Identify the coefficient of [tex]\( x \)[/tex]:
We can see that in the expression [tex]\( x^2 + Px - 30 \)[/tex], [tex]\( P \)[/tex] is the coefficient of [tex]\( x \)[/tex]. By comparing it to [tex]\( x^2 - 13x - 30 \)[/tex], we can see that the coefficient of [tex]\( x \)[/tex] is [tex]\(-13\)[/tex].
So, the value of [tex]\( P \)[/tex] is:
[tex]\[ P = -13 \][/tex]
