Solve the equation x^3-17 x+4=0 given that 4 is a root. |...

Name: Problem 48, Chapter 3: Precalculus
Uploaded: 2020-03-09T19:28:51-08:00
Duration: 4 min 37 s
Channel: Thomas Emment
Description: Solve the equation x^3-17 x+4=0 given that 4 is a root.

Key Concepts

Quadratic Formula

The quadratic formula provides a systematic way to find the roots of a quadratic equation, which is especially useful once a cubic equation has been reduced to a quadratic factor. It identifies solutions by substituting the coefficients of the quadratic expression into the formula.

Factor Theorem

The factor theorem states that if a polynomial f(x) equals zero for a particular value, say a (i.e., f(a)=0), then (x - a) is a factor of the polynomial. This theorem is crucial for breaking down higher-degree polynomials into simpler, solvable factors.

Synthetic Division

Synthetic division is a streamlined method of dividing polynomials when one of the divisors is linear. It simplifies the process of factoring by reducing a polynomial using a known factor, making it easier to find other roots.

Cubic Equation Solving

Solving cubic equations involves finding the roots of a third-degree polynomial. When a single root is known, the equation can often be factored into a product of a linear factor and a quadratic factor, ultimately simplifying the process to find all solutions.

Transcript

00:01 Okay, okay, we want to solve this equation and we're told that 4 is a root.

00:06 That means that x minus 4 is going to be a factor of this.

00:13 So if we perform long division or however, whichever division method you prefer, i'm more of a fan of long division myself.

00:20 So if we perform long division, we should get a nice simple squared equation, which you can either put into the quadratic formula, or hopefully we'll be able to just see for ourselves what the other roots are.

00:32 Yeah, so we've got x cubed minus 17x plus four.

00:38 Now, x goes into x cubed x squared x squared times.

00:41 So we'll have minus x cubed minus 4x squared.

00:47 Minus minus, well, obviously it's going to be zero, so minus minus minus be plus.

00:51 So we'll have 4x squared and bring down the rest of our equation.

00:56 17x plus 4.

00:58 Okay, how many x can we get into 4x squared? well, so it's going to be 4x.

01:03 So, minus 4x squared, minus 16x.

01:10 And if we minus minus and plus, so it's simply going to be left with x plus 4, which is nice.

01:18 Wait, should we? sorry, minus x plus 4.

01:21 Almost fuse yourself for a minute.

01:23 But do be able to catch yourself on this, because this should come out of zero because you've been told that 4 is a root.

01:32 So if you do make a mistake here, you know that it's because you haven't got a 0 here.

01:36 So we're just going to have one lot of the last one.

01:38 I hope you can see minus 1 lot of this.

01:42 We'll get rid of this last equation here.

01:44 Minus 1 times 1 .1.

01:45 So we're left with x squared plus 4x minus 1.

01:54 And looking at that, i actually can't find the roots from top of my head.

01:58 So we're going to have to use a quadratic formula...

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