Math / Algebra

QuestionThe graph of a polynomial $f(x)=(5 x-3)(x+4)(x+a)$ has $x$ -intercepts at $-4, \frac{3}{5}$ , and 6 . What is the value of $a$ ?

Studdy Solution

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!

STEP 11

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

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QuestionThe graph of a polynomial $f(x)=(5 x-3)(x+4)(x+a)$ has $x$ -intercepts at $-4, \frac{3}{5}$ , and 6 . What is the value of $a$ ?

Studdy Solution

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!

STEP 11

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

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QuestionThe graph of a polynomial f(x)=(5x−3)(x+4)(x+a)f(x)=(5 x-3)(x+4)(x+a)f(x)=(5x−3)(x+4)(x+a) has xxx-intercepts at −4,35-4, \frac{3}{5}−4,53​, and 6 . What is the value of aaa ?

Studdy Solution

STEP 1

What is this asking? We're given a polynomial in factored form with unknown aaa, and we know where the graph crosses the x-axis.We need to find the value of aaa. 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)(x+6)(x+6)!

STEP 2

1. Relate x-intercepts to factors2. Solve for *a*

STEP 3

Alright, so we're given this nicely factored polynomial f(x)=(5x−3)(x+4)(x+a)f(x) = (5x - 3)(x + 4)(x + a)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)=0f(x) = 0f(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=−4x = -4x=−4, x=35x = \frac{3}{5}x=53​, and x=6x = 6x=6.Let's see how these relate to our factors.

STEP 6

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

STEP 7

If x=35x = \frac{3}{5}x=53​, then the factor (5x−3)(5x - 3)(5x−3) becomes 5⋅35−3=3−3=05 \cdot \frac{3}{5} - 3 = 3 - 3 = 05⋅53​−3=3−3=0.Awesome! Another match!

STEP 8

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

STEP 9

So, if x=6x = 6x=6, we plug that into (x+a)(x + a)(x+a) to get 6+a=06 + a = 06+a=0.

STEP 10

To **isolate** aaa, 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 aaa on the left side.On the right side, 000 plus negative 6 is negative 6.So, a=−6a = -6a=−6.Boom!

STEP 11

The value of aaa is −6-6−6.

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Start learning now

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QuestionThe graph of a polynomial $f(x)=(5 x-3)(x+4)(x+a)$ has $x$ -intercepts at $-4, \frac{3}{5}$ , and 6 . What is the value of $a$ ?

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 $(x+6)$ !

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

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!

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!

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.

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

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

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

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

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 $-6$ .