Math  /  Algebra

QuestionConsider the following polynomial function. f(x)=x46x3+x2+42x56f(x)=x^{4}-6 x^{3}+x^{2}+42 x-56
Step 3 of 4 : Find the xx-intercept(s) at which ff crosses the axis. Express the intercept(s) as ordered pair(s).
Answer
Select the number of xx-intercept(s) at which ff crosses the axis.
Selecting an option will display any text boxes needed to complete your answer. none 1 2 3

Studdy Solution

STEP 1

What is this asking? We're looking for the x-intercepts where our funky function *crosses* the x-axis, not just touches it! Watch out! An x-intercept *touches* the x-axis when the function just grazes it, but we want where it *crosses* through!

STEP 2

1. Find the roots
2. Determine crossing behavior

STEP 3

We're given the polynomial f(x)=x46x3+x2+42x56f(x) = x^4 - 6x^3 + x^2 + 42x - 56.
Let's **factor** this bad boy to find its roots!
By trying some small integer values, we find that f(2)=0f(2) = 0, so (x2)(x-2) is a factor.
Dividing our polynomial by (x2)(x-2) gives us f(x)=(x2)(x34x27x+28)f(x) = (x-2)(x^3 - 4x^2 - 7x + 28).

STEP 4

Now, let's **factor** the cubic term.
Notice that x=2x=2 is a root again!
Dividing the cubic by (x2)(x-2) gives us x34x27x+28=(x2)(x22x14)x^3 - 4x^2 - 7x + 28 = (x-2)(x^2 - 2x - 14).
So, f(x)=(x2)2(x22x14)f(x) = (x-2)^2(x^2 - 2x - 14).

STEP 5

We have f(x)=(x2)2(x22x14)f(x) = (x-2)^2(x^2 - 2x - 14).
The first factor gives us a **root** x=2x=2.
For the quadratic factor, we use the **quadratic formula**: x=b±b24ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
Here, a=1a=1, b=2b=-2, and c=14c=-14, so x=2±(2)24(1)(14)2(1)=2±4+562=2±602=2±2152=1±15x = \frac{2 \pm \sqrt{(-2)^2 - 4(1)(-14)}}{2(1)} = \frac{2 \pm \sqrt{4 + 56}}{2} = \frac{2 \pm \sqrt{60}}{2} = \frac{2 \pm 2\sqrt{15}}{2} = 1 \pm \sqrt{15}.
So, our **roots** are x=2x=2, x=1+15x=1+\sqrt{15}, and x=115x=1-\sqrt{15}.

STEP 6

Remember, a **root** with an *odd* multiplicity *crosses* the x-axis, while a root with an *even* multiplicity just *touches* it.
Our **root** x=2x=2 comes from (x2)2(x-2)^2, so it has multiplicity **2** (even!), meaning it just *touches* the x-axis.
The other two **roots**, 1+151+\sqrt{15} and 1151-\sqrt{15}, each have multiplicity **1** (odd!), so they *cross* the x-axis!

STEP 7

The **roots** where the function *crosses* the x-axis are 1+151+\sqrt{15} and 1151-\sqrt{15}.
As ordered pairs, these are (1+15,0)(1+\sqrt{15}, 0) and (115,0)(1-\sqrt{15}, 0).

STEP 8

There are *two* x-intercepts where f(x)f(x) crosses the x-axis: (1+15,0)(1+\sqrt{15}, 0) and (115,0)(1-\sqrt{15}, 0).

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