The key idea
The remainder when p(x) is divided by x−a is p(a). So x−a is a factor exactly when p(a)=0. Each known remainder or factor gives one equation for the unknown coefficients.
The trap
Plugging in x=−a for the divisor x−a.
Can Desmos do it?
Desmos evaluates p(a) in a blink, but unknown coefficients still need equations from you.
College Board skill: Equivalent expressions · Advanced Math
One question from each level. The drill has 12 at every level, and moves you up after four right in a row.
Level 1 · Warm-up
What is the remainder when p(x)=3x3+3x2+6x−6 is divided by x−2?
Show the answer
Answer: 42By the remainder theorem, the remainder is p(2)=42.
Level 2 · Test level
The polynomial p(x)=x3+x2+kx+7, where k is a constant, has x+1 as a factor. What is the value of k?
Show the answer
Answer: 7Since x+1 is a factor, p(−1)=0: −1+1−k+7=0, so −k=−7 and k=7.
Level 3 · Beyond the test
When p(x)=x3+ax2+bx−3 is divided by x−2, the remainder is −1. When p(x) is divided by x+3, the remainder is −81. What is the value of a+b?
Show the answer
Answer: 1The remainder theorem gives p(2)=−1 and p(−3)=−81: 8+4a+2b−3=−1 and −27+9a−3b−3=−81. Solving: a=−4 and b=5, so a+b=1.
Practice all 36 questions
Answer in the drill to get instant feedback, climb the levels, and feed your misses into Tailored practice. Free, no account.
The discriminant: how many solutionsCounting solutions and finding tangent lines with b² − 4ac.36 questions, 3 levelsVertex form, maximum and minimumMaximums, minimums and symmetry, including projectiles.36 questions, 3 levelsSum and product of rootsSums, products and missing roots without solving.36 questions, 3 levelsExponential growth and decayReading growth factors and converting between time periods.36 questions, 3 levelsExponent rules and radicalsFractional exponents, radicals and mixed bases.36 questions, 3 levelsShifts and stretches of graphsShifts, stretches and where the zeros and vertex move.36 questions, 3 levelsRadical and rational equationsSquare roots and fractions that create fake solutions.36 questions, 3 levels