Answer: $ax^2 + bx + c$

Definition of a quadratic expression: A quadratic expression is a polynomial of degree 2, written in the form $ax^2 + bx + c$ where $a \neq 0$. Why the correct answer works: $ax^2 + bx + c$ has a highest power of 2 on the variable, which defines a quadratic. Why distractors fail: Option A ($ax + b$) is linear (degree 1). Option C ($ax^3 + bx^2 + cx + d$) is cubic (degree 3). Option D ($a/x^2 + b$) involves a negative exponent and is not a polynomial.

Answer: $7x^2$

Check each expression's degree and form: A quadratic expression is a polynomial whose highest exponent on the variable is exactly 2. Why the correct answer works: $7x^2$ is a monomial with degree 2, which qualifies as a quadratic expression. Why distractors fail: $2x^{-2} + x$ contains a negative exponent and is not a polynomial. $x^2 + x^3$ has a highest exponent of 3, making it cubic. $8x + 12$ has degree 1, making it linear.

Answer: $(x - 3)(x + 3)$

Expand each expression: $5(x+2) = 5x + 10$ (degree 1). $(x-3)(x+3) = x^2 - 9$ (degree 2). $x(x^2-1) = x^3 - x$ (degree 3). $2(3x+4)+x = 7x + 8$ (degree 1). Why the correct answer works: $(x-3)(x+3)$ expands to $x^2 - 9$, a degree 2 polynomial. It is a quadratic in disguise because it does not initially appear in $ax^2 + bx + c$ form. Why distractors fail: Options A and D simplify to linear expressions. Option C simplifies to a cubic expression.

Answer: The student is incorrect; a quadratic is defined by degree 2, not by the number of terms

Definition of quadratic: A quadratic expression is any polynomial of degree 2. The number of terms (monomial, binomial, or trinomial) does not affect whether it is quadratic. Why the correct answer works: $6x^2$ has degree 2, so it is quadratic regardless of having just one term. Why distractors fail: Option A incorrectly ties quadratic status to three terms. Option B wrongly calls a monomial with a variable a constant. Option D overgeneralizes—$5x$ is a single-term expression but not quadratic.

Answer: Add the degrees of the two factors: degree 1 + degree 1 = degree 2

Key principle: multiplying polynomials: When two polynomials are multiplied, the degree of the product equals the sum of the degrees of the factors. Apply to this problem: $(2x+1)$ has degree 1 and $(x-4)$ has degree 1. The product has degree $1 + 1 = 2$, confirming it is quadratic. Why distractors fail: Option A: term count does not determine degree. Option C: multiplying constants gives the constant term of the product, not the degree. Option D: having the same variable is necessary but does not determine degree.

Answer: False, because a quadratic can be a binomial such as $x^2 + 5$ or a monomial such as $3x^2$

Understand what defines a quadratic: A quadratic expression is defined by its degree (the highest exponent is 2), not by the number of terms it contains. Why the correct answer works: When $b = 0$ or $c = 0$ (or both), the expression still has degree 2. For example, $3x^2$ is a monomial quadratic and $x^2 + 5$ is a binomial quadratic. Why distractors fail: Option A incorrectly assumes all three coefficients must be nonzero. Option C is wrong because setting $b$ or $c$ to zero does not change the highest exponent. Option D is wrong because a polynomial of degree 2 cannot have more than three terms with distinct powers of $x$.

Answer: $5$, then $3x - 2$, then $x^2 + 1$, then $x^3 + x$

Determine each degree: $5$ is a constant (degree 0). $3x - 2$ has degree 1. $x^2 + 1$ has degree 2. $x^3 + x$ has degree 3. Why the correct answer works: Option C lists all four expressions in correctly ascending order of degree: degree 0, then degree 1, then degree 2, then degree 3. Why distractors fail: Option A places degree 2 before degree 1. Option B places degree 2 before degree 1. Option D places degree 1 before degree 0 and degree 3 before degree 2.

Answer: $x^3 - x^3 + 2x^2 - x$

Simplify each expression: Option A: $x^3 - x^3 + 2x^2 - x = 2x^2 - x$ (degree 2). Option B: $x^2 - x^2 + 3x + 1 = 3x + 1$ (degree 1). Option C: $2x(0 + 5) = 10x$ (degree 1). Option D: $x^3 + x^2 - x^3 + x^3 = x^3 + x^2$ (degree 3). Why the correct answer works: Option A initially appears to be cubic because of the $x^3$ terms, but they cancel, leaving $2x^2 - x$, which is quadratic. Why distractors fail: Option B simplifies to linear. Option C simplifies to linear. Option D simplifies to cubic because only one $x^3$ term cancels.

Answer: A quadratic has a highest variable exponent of 2, whereas a linear expression has a highest variable exponent of 1

Key distinction: degree: The fundamental difference is the degree of the polynomial. A quadratic has degree 2 (highest exponent is 2) and a linear expression has degree 1 (highest exponent is 1). Why the correct answer works: Option C correctly identifies that the highest power of the variable is 2 for quadratics and 1 for linear expressions. Why distractors fail: Option A is wrong because coefficient size is unrelated to expression type. Option B is wrong because both types can have varying numbers of terms. Option D is wrong because any variable name can be used in either type.

Answer: (i) only

Determine the degree of each expression: (i) $2t^2 - 3t + 1$: highest exponent is 2 → degree 2 (quadratic). (ii) $4t - 9$: highest exponent is 1 → degree 1 (linear). (iii) $t^3 + t^2$: highest exponent is 3 → degree 3 (cubic). Why the correct answer works: Only expression (i) has degree 2, making it the only quadratic. Why distractors fail: Option A: (ii) is linear. Option B: (iii) is cubic. Option D: (iii) is cubic despite containing a $t^2$ term—the highest exponent determines the degree.

Answer: $x^3 + x^2 - x^3 + 5$

Check each expression: Option A: $x^3 + x^2 - x^3 + 5 = x^2 + 5$ (degree 2, 2 terms — quadratic binomial). Option B: $2x^3 - x^3 + x + 3 = x^3 + x + 3$ (still cubic). Option C: $x^3 - x^3 + 4x + 2 = 4x + 2$ (linear, not quadratic). Option D: $x^3 - x^3 + x^2 + x + 1 = x^2 + x + 1$ (quadratic trinomial, not binomial). Why the correct answer works: Option A has $x^3$ terms that cancel, leaving $x^2 + 5$—exactly a quadratic binomial. Before simplification, the $x^3$ term makes it look cubic. Why distractors fail: Option B remains cubic. Option C becomes linear. Option D becomes a quadratic trinomial (three terms), not a binomial.

Answer: $x(x + 4)$

Expand each expression: $x(x+4) = x^2 + 4x$ (degree 2). $x(x^2+1) = x^3 + x$ (degree 3). $3(2x-5) = 6x - 15$ (degree 1). $x^2(x+2) = x^3 + 2x^2$ (degree 3). Why the correct answer works: $x(x+4)$ expands to $x^2 + 4x$, a polynomial of degree 2, which is quadratic. Why distractors fail: Options B and D both produce degree 3 expressions after expansion. Option C produces a linear (degree 1) expression.

Answer: The reasoning is flawed; after simplification, the $x^2$ terms cancel and the result is linear, so the expression is not quadratic

Expand and simplify: $(x+2)^2 - x^2 = x^2 + 4x + 4 - x^2 = 4x + 4$. The $x^2$ terms cancel completely. Determine the degree of the simplified expression: $4x + 4$ has degree 1, making the expression linear, not quadratic. Why distractors fail: Options A and D incorrectly assume the presence of a squared factor guarantees a quadratic result. Option C confuses the number of terms with degree.

Answer: The claim is incorrect; expanding $(x+5)^2$ yields $x^2 + 10x + 25$, which is quadratic

Recognize a quadratic in disguise: An expression does not need to be written in standard form to be quadratic. If it simplifies to a degree 2 polynomial, it is quadratic. Expand to verify: $(x+5)^2 = x^2 + 10x + 25$. The highest power is 2, confirming it is quadratic. Why distractors fail: Option A wrongly restricts quadratics to standard form only. Option C invents a false category. Option D is absurd—parentheses alone do not determine degree.

Answer: 2

Recall the definition of degree: The degree of a polynomial is the highest exponent on the variable in the expression. Apply to this expression: In $3x^2 + 5x - 7$, the highest exponent on $x$ is 2, so the degree is 2. Why distractors fail: Option A (1) is the exponent of the second term, not the highest. Option C (3) is the coefficient of the leading term, not the exponent. Option D (7) is the constant term's absolute value.

Answer: $-2x^2 + 9$

Check the degree of each expression: A quadratic expression must be a polynomial with the highest exponent on the variable equal to 2. Why the correct answer works: $-2x^2 + 9$ has a highest power of 2, making it a quadratic expression (specifically a binomial quadratic). Why distractors fail: $5x + 2$ is degree 1 (linear). $4x^3 - x$ is degree 3 (cubic). $\sqrt{x} + 3$ involves a fractional exponent ($x^{1/2}$) and is not a polynomial.

Answer: $k = 0$

Recall the requirement for a quadratic: In $ax^2 + bx + c$, the expression is quadratic only if $a \neq 0$. If $a = 0$, the $x^2$ term vanishes and the expression reduces to a lower degree. Apply to this problem: When $k = 0$, the expression becomes $3x - 7$, which is degree 1 (linear), not quadratic. Why distractors fail: For $k = -1$, $k = 1$, or $k = 7$, the $x^2$ term remains, so the expression stays quadratic. The sign or magnitude of $k$ does not matter as long as $k \neq 0$.

Answer: No, because expanding gives $(a+b)x + ab$, which is linear when $a + b \neq 0$ or constant when $a + b = 0$—either way, not quadratic

Expand the full expression: $(x+a)(x+b) = x^2 + (a+b)x + ab$. Subtracting $x^2$ gives $x^2 + (a+b)x + ab - x^2 = (a+b)x + ab$. Determine the degree: If $a + b \neq 0$, the result $(a+b)x + ab$ has degree 1 (linear). If $a + b = 0$ (for example, $a = 3$ and $b = -3$), the result is just the constant $ab$, which has degree 0. In neither case is the expression quadratic. Why distractors fail: Option A ignores the subtraction of $x^2$ that cancels the quadratic term. Option B notes the presence of $x^2$ during expansion but ignores that it cancels. Option D is wrong because $a$ and $b$ are constants, not additional variables.