Math Snap

STEP 1

What is this asking?
We're given a polynomial in factored form with unknown $a$, and we know where the graph crosses the x-axis.
We need to find the value of $a$.
Watch out!
Don't mix up the x-intercepts with the factors.
Remember, an x-intercept of 6 doesn't mean the factor is $(x+6)$!

STEP 2

1. Relate x-intercepts to factors
2. Solve for a

STEP 3

Alright, so we're given this nicely factored polynomial $f(x) = (5x - 3)(x + 4)(x + a)$.
When a polynomial is factored like this, it's super easy to find the x-intercepts!

STEP 4

Remember, an x-intercept is where the graph of the function crosses the x-axis.
This happens when $f(x) = 0$.
Since we're multiplying factors together, the whole thing equals zero if any of the factors are zero!

STEP 5

We're given that the x-intercepts are at $x = -4$, $x = \frac{3}{5}$, and $x = 6$.
Let's see how these relate to our factors.

STEP 6

If $x = -4$, then the factor $(x+4)$ becomes $(-4) + 4 = 0$.
Perfect! This matches the given polynomial.

STEP 7

If $x = \frac{3}{5}$, then the factor $(5x - 3)$ becomes $5 \cdot \frac{3}{5} - 3 = 3 - 3 = 0$.
Awesome! Another match!

STEP 8

Now, we've got this third x-intercept at $x = 6$.
We know that this must make the factor $(x + a)$ equal to zero.

STEP 9

So, if $x = 6$, we plug that into $(x + a)$ to get $6 + a = 0$.

STEP 10

To isolate $a$, we want to add the opposite of 6 to both sides of the equation.
Since 6 plus negative 6 is zero, we're left with $a$ on the left side.
On the right side, $0$ plus negative 6 is negative 6.
So, $a = -6$.
Boom!

The value of $a$ is $-6$.