1 of 8 0-day streak

Bracket and approximate a polynomial root by bisection

Question 1 of 8

No calculator

Consider the polynomial function \(P(x)\)

P(x)=6x^{2} + 44x + 70

The initial interval is \([-3,-2]\)

The endpoint values are non-zero. Use exact values while completing the bisection table

Use 3 successive bisection rows, then report the final retained interval's midpoint rounded to 1 decimal place

Use nearest rounding, with exact halfway ties away from zero. The answer requested is this fixed-step midpoint estimate, not a certificate of the true root's rounding

(a)2 marks

Evaluate the two initial endpoints exactly

\(P(-3)\) \(P(-2)\)

(b)1 mark

Select the conclusion justified by the endpoint signs

Because P(-3) and P(-2) have opposite signs, the interval contains at least one real root

Because P(-3) and P(-2) have opposite signs, the interval contains exactly one real root

Because neither endpoint is a zero, the interval contains no real root

Complete the exact bisection table. For every row, give the current endpoints, their midpoint and the value of \(P\) at the midpoint

\(L_1\) \(R_1\) \(M_1\) \(P(M_1)\) \(L_2\) \(R_2\) \(M_2\) \(P(M_2)\) \(L_3\) \(R_3\) \(M_3\) \(P(M_3)\)

(d)1 mark

After row 1, choose the half-interval to retain

[-\frac{5}{2},-2]

[-3,-\frac{5}{2}]

(e)1 mark

After row 2, choose the half-interval to retain

[-\frac{9}{4},-2]

[-\frac{5}{2},-\frac{9}{4}]

(f)1 mark

After row 3, choose the half-interval to retain

[-\frac{5}{2},-\frac{19}{8}]

[-\frac{19}{8},-\frac{9}{4}]

(g)1 mark

Round the midpoint of the interval retained after row 3 to 1 decimal place. Enter the rounded value with the stated decimal precision

\(M\) Format: 1 d.p.