QCM : Mastering Polynomial Inequalities and Graphs — 24 questions

Explication

Adding the same real number to both sides preserves the inequality direction. Reversal happens when multiplying or dividing by a negative number, not when adding.

Explication

Multiplying by a negative number reverses the order of the two sides, so the inequality sign flips. This is one of the basic sign rules for inequalities.

Explication

A product is positive when both factors have the same sign, which gives x<0 or x>4. The endpoints are excluded because the inequality is strict.

Explication

For non-strict inequalities, points where the graph equals zero must be included. Touching the x-axis means the expression is exactly 0 at that x-value.

Explication

The expression is undefined when the denominator is 0, so x=2 must be excluded. That point cannot be part of the solution set.

Explication

A squared factor is non-negative and, away from the excluded value, positive, so it does not change the inequality direction. This is used to avoid case-splitting based on sign.

Explication

The radicand must be non-negative, so 1-x^2≥0, which gives -1≤x≤1. Outside that interval, the square root is not defined for real numbers.

Explication

The graphs intersect at x=0 and x=1, and the stated solution includes x=0. This comes from comparing the absolute-value graph with the square-root graph on the restricted domain.

Explication

The inequality x^2+y^2≤9 represents the inside of the circle centered at the origin with radius 3, including the boundary. A strict inequality would exclude the boundary.

Explication

Setting x^2=2x+3 gives x^2-2x-3=0, which factors to (x-3)(x+1)=0. The corresponding points are (-1,1) and (3,9).

Explication

The degree is the highest power of x with a nonzero coefficient, which is 3. The constant term 7 does not affect the degree.

Explication

A monic polynomial has leading coefficient 1, and x^4-3x^2+1 fits that definition. The other choices have leading coefficients different from 1 or are not written as a polynomial of positive degree.

Explication

When the divisor is linear, the remainder must have degree less than 1, so it is a constant. It is not necessarily zero unless the divisor is a factor.

Explication

By the remainder theorem, the remainder on division by (x-a) equals P(a). So P(3)=5.

Explication

The factor theorem states that P(a)=0 if and only if (x-a) is a factor of P(x). This is the key link between zeros and factors.

Explication

A zero remainder means the division is exact, so the divisor is a factor. By the factor theorem, 4 is then a root of the polynomial.

Explication

For a quadratic, the sum of the roots equals -b/a. The product of the roots is c/a, not the sum.

Explication

For a cubic, the product of the roots is -d/a. The coefficient c/a corresponds to the sum of pairwise products instead.

Explication

Each differentiation lowers the multiplicity by 1, so a triple root becomes a double root in the first derivative. This matches the rule r, r-1, r-2, and so on.

Explication

A multiple root must also be a root of the derivative because differentiation reduces multiplicity by one. So if c is repeated, both P(c)=0 and P'(c)=0 hold.

Explication

A zero has multiplicity r when the factor (x-c) is repeated r times in the factorization. This is what distinguishes a multiple root from a simple root.

Explication

Differentiating reduces the multiplicity by one, so a root of multiplicity r in P becomes multiplicity r-1 in P'. This is why a multiple root of P must also be a root of P'.

Explication

A polynomial function is defined for every real x, and its graph is continuous and differentiable everywhere. Unlike a general equation, it does not have gaps or breaks.

Explication

Odd-degree polynomial graphs have opposite end behavior, so one end rises while the other falls. Same-direction end behavior is a feature of even-degree polynomials.