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Chapter 7: Problem 17

Factorise \(\nu^{3}-3 \nu^{2}-33 \nu+35\) given that \((\nu-7)\) is a factor.

Short Answer

Expert verified
Question: Completely factorize the following polynomial: \(\nu^{3}-3 \nu^{2}-33 \nu+35\) given that \((\nu-7)\) is a factor. Answer: \(\nu^{3}-3 \nu^{2}-33 \nu+35 = (\nu-7)(\nu + 5)(\nu - 1)\).

Step by step solution

01

Perform synthetic division with the given factor

We are given that \((\nu-7)\) is a factor of the polynomial. We can use synthetic division to divide the polynomial by \((\nu-7)\). Here's how to set up the synthetic division: ``` 7 | 1 -3 -33 35 7 28 -35 ------------------- 1 4 -5 0 ``` The result of the synthetic division is a quadratic polynomial: \(\nu^{2} + 4 \nu - 5\). So, \(\nu^{3}-3 \nu^{2}-33 \nu+35 = (\nu-7)(\nu^{2} + 4 \nu - 5)\).
02

Factor the quadratic polynomial

Now, we need to factor the quadratic polynomial \(\nu^{2} + 4 \nu - 5\). We will do this by finding two numbers that multiply to \(-5\) and add to \(4\). The two numbers are \(5\) and \(-1\). Thus, we can factor the quadratic as follows: \(\nu^{2} + 4\nu -5 =(\nu + 5)(\nu - 1)\). Now the original polynomial is completely factored: \(\nu^{3}-3 \nu^{2}-33 \nu+35 = (\nu-7)(\nu + 5)(\nu - 1)\).

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Most popular questions from this chapter

(a) Write down the general form of a linear equation. (b) Explain what is meant by the root of a linear equation.

(a) If \(3 x^{2}+13 x+4=(x+4) \times\) a polynomial, state the degree of the polynomial. (b) What is the coefficient of \(x\) in this unknown polynomial?

Table \(7.3\) shows the values of \(x\) and \(y\). Given that \(y\) is proportional to \(x\) (a) find an equation connecting \(y\) and \(x\) (b) calculate the value of \(y\) when \(x=36\) (c) calculate the value of \(x\) when \(y=200\) $$ \begin{array}{lclllc} \hline x & 5 & 10 & 15 & 20 & 25 \\ y & 22.5 & 45 & 67.5 & 90 & 112.5 \\ \hline \end{array} $$

Verify that the given value is a solution of the given equation. $$ \frac{1}{3} x+\frac{4}{3}=2, x=2 $$

Solve the simultaneous equations $$ 3 x-2 y=11,5 x+7 y=39 $$
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