1 of 8 0-day streak

Bracket and approximate a polynomial root by bisection

Question 1 of 8

No calculator

A calibration model has signed output \(P(x)\), where \(x\) is a setting. A zero of \(P\) is a setting at which the output is zero

P(x)=10x^{2} - 54x + 68

The initial interval is \([3,4]\)

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

Refine the bracket for exactly 3 bisection rows. Round the midpoint of the resulting interval 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

\(P(3)\) \(P(4)\)

(b)1 mark

Select the conclusion justified by the endpoint signs

Because P(3) and P(4) have opposite signs, the interval contains exactly one real root

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

Because P(3) and P(4) have opposite signs, the interval contains at least one 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{7}{2},4]

[3,\frac{7}{2}]

(e)1 mark

After row 2, choose the half-interval to retain

[\frac{13}{4},\frac{7}{2}]

[3,\frac{13}{4}]

(f)1 mark

After row 3, choose the half-interval to retain

[\frac{27}{8},\frac{7}{2}]

[\frac{13}{4},\frac{27}{8}]

(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.