Problem 73 - Precalculus Mathematics for Calculus

Question

Show that the given values for \(a\) and \(b\) are lower and upper bounds for the real zeros of the polynomial. $$P(x)=8 x^{3}+10 x^{2}-39 x+9 ; \quad a=-3, b=2$$

Step-by-Step Solution

Answer

-3 is a lower bound and 2 is an upper bound for the polynomial's zeros.

Step 1: Understand Bounds and Zeros

A lower bound "a" means that there are no real zeros of the polynomial less than "a". An upper bound "b" means that there are no real zeros greater than "b". We need to show that within interval \([-3, 2]\), all zeros, if they exist, are contained.

Step 2: Check lower bound using synthetic division

Apply synthetic division on \(P(x)\) using \(a = -3\). If full division results in remainder position having opposite signs, then \(-3\) is a lower bound for zeros.\[\begin{array}{r|rrrr}-3 & 8 & 10 & -39 & 9\ & & -24 & 42 & -9 \\hline & 8 & -14 & 3 & 0 \\end{array}\] Since the signs do not change in the remainder and quotient (\([8, -14, 3, 0]\)), \(-3\) is indeed a lower bound.

Step 3: Check upper bound using synthetic division

Apply synthetic division on \(P(x)\) using \(b = 2\). If division results in the last remainder being positive, then 2 is an upper bound.\[\begin{array}{r|rrrr}2 & 8 & 10 & -39 & 9 \ & & 16 & 52 & 26 \ \hline & 8 & 26 & 13 & 35 \ \end{array}\] The last value is 35 which is positive, hence, confirming that \(2\) is an upper bound.

Step 4: Conclusion

Since no sign change in synthetic division for \(a=-3\) and a positive final remainder for \(b=2\), these values correctly bound the real zeros. Hence, \(-3\) and \(2\) are lower and upper bounds for the zeros of \(P(x)\).

Key Concepts

Synthetic Division

Synthetic division is a method used to divide polynomials more efficiently than regular polynomial division, especially when dividing by a linear term of the form \(x - c\).
It's a bit like a shortcut and is particularly useful when checking for upper and lower bounds of polynomial zeros. In synthetic division, coefficients of the polynomial are written in a row. You substitute the zero value, \(c\), to test in the small box on the left – not unlike a long division process.
Then you perform a series of multiplications and additions:

• Multiply the number in the small box by the initial polynomial coefficient.
• Add the result to the next coefficient and continue to move across.

The outcome tells you whether the divisor is effectively a factor or not. If it is, the last remainder will be zero.
If not, it can still provide valuable clues about bounds and zeros.

Polynomial Zeros

Zeros of a polynomial, also known as roots, are the values of \(x\) that make the polynomial equal to zero.
Finding the zeros of a polynomial involves solving the equation \(P(x) = 0\). Understanding zeros is essential because:

• They indicate where the graph of the polynomial crosses or touches the x-axis.
• They can be real or complex numbers.

When working with synthetic division, zeros have a key role. They help us verify whether the result of the division can confirm our boundary conditions.
By calculating the polynomial at different potential zero locations, and carrying out techniques like synthetic division, you can confirm whether your polynomial has real zeros and even ascertain their location.

Upper and Lower Bounds

Upper and lower bounds are estimates that encompass all real zeros of a polynomial.
The goal is to find such bounds so we know where real zeros can and cannot be. To find these using synthetic division:

• For a lower bound, test if all coefficients (or results) are non-positive.
• For an upper bound, test whether the final remainder is positive.

The signs of the results obtained from performing synthetic division on a polynomial gives insights:

• A change or no change in sign of the result determines whether a test bound indeed is a valid boundary.

Knowing these bounds helps in accurately narrowing down the regions where the zeros of the polynomial may lie, thus aiding in deeper analysis or more refined graphing.