Linear polynomials

Correct answer: A. The sign of the constant term has been handled incorrectly; the correct zero is \(-5\).

Explanation: For a zero, set \(p(x)=0\): \(3x+15=0\Rightarrow3x=-15\Rightarrow x=-5\). The student dropped the negative sign after division. Exam tip: substitute the value back to verify that the polynomial becomes zero.

Correct answer: A. x-65

Explanation: Here, 2p(x)=2(8x-5)=16x-10 and 5q(x)=5(3x+11)=15x+55. Therefore, 2p(x)-5q(x)=(16x-10)-(15x+55)=16x-10-15x-55=x-65. The option x+45 results from not applying the negative sign correctly to the constant term of q(x). Exam tip: When subtracting a bracket, change the sign of every term inside it.

Correct answer: A. \(a=5\)

Explanation: If \(x=-2\) is a zero of \((4a-5)x+30\), the polynomial must equal zero at \(x=-2\): \((4a-5)(-2)+30=0\). Thus, \(-8a+10+30=0\), or \(-8a+40=0\), giving \(a=5\). For \(a=\frac{5}{4}\), the coefficient of \(x\) becomes zero and the expression is \(30\), so it cannot have a zero. Exam tip: substitute the given zero for \(x\) and equate the polynomial to zero.

Correct answer: C. 28

Explanation: Since 7 is a zero, p(7)=0. Therefore, 7a+b=0. Substituting a=-4 gives 7(-4)+b=0, or -28+b=0. Hence, b=28. The close distractor -28 is the value of 7a, not of b. Exam tip: if r is a zero of the linear polynomial ax+b, use ar+b=0 to find an unknown coefficient.

Correct answer: B. Every real value of \(c\)

Explanation: \(p(4)=44+c\) and \(p(1)=11+c\). Hence, \(p(1)+33=11+c+33=44+c=p(4)\). The term \(c\) occurs equally on both sides, so it cancels for every real value of \(c\). Therefore, every real value of \(c\) is valid. \(c=0\) and \(c=33\) are only particular values, not necessary conditions. Exam tip: Evaluate the polynomial at the given inputs first, then simplify the constant terms.

Correct answer: C. (13x+9)

Explanation: For a linear polynomial ax+b, the zero is found by setting the expression equal to zero and solving for x. The required value is \(-\frac{9}{13}\), so the equation should lead to 13x=-9. A positive constant term 9 together with coefficient 13 produces exactly this result. The sign is important: changing the sign of the constant changes the sign of the zero.

In option C, solve 13x+9=0. Subtracting 9 gives 13x=-9, and division by 13 gives x=-\frac{9}{13}. Therefore option C has the required zero. Option A gives a positive zero \(\frac{9}{13}\), while options B and D lead to different fractions, so they do not satisfy the condition.

Correct answer: C. (x=30)

Explanation: From (\frac{8}{15}x-16=0), (\frac{8}{15}x=16) and (x=30). With a fractional coefficient, multiply by the reciprocal.

Correct answer: B. 2

Explanation: Both p(x) and q(x) are linear polynomials, so each has degree 1. Their product is (x-10)(x+3)=x²-7x-30. The highest power of x is 2, so the degree of the product is 2. Option 1 is incorrect because it is the degree of each individual polynomial, not of their product. Exam tip: For non-zero polynomials, the degree of a product is usually the sum of their degrees.

Correct answer: A. \(9x+10\)

Explanation: Given \(p(x)=9x-17\), replace the entire \(x\) with \(x+3\): \(p(x+3)=9(x+3)-17=9x+27-17=9x+10\). Hence, \(9x+10\) is correct. \(9x+27\) results from forgetting to subtract 17. Exam tip: always put the substituted expression in brackets when evaluating a function.

Correct answer: B. \(19-14x\)

Explanation: Given \(p(x)=19-7x\), replace the input \(x\) by the entire expression \(2x\): \(p(2x)=19-7(2x)=19-14x\). Hence, option B is correct. \(38-14x\) would result from incorrectly doubling the constant term 19 as well; substitution changes only the input. Exam tip: In \(p(2x)\), replace every \(x\) in the rule by \(2x\).

Correct answer: A. \(\sqrt{2}x-5\)

Explanation: A polynomial may have irrational real coefficients. In option A, the highest power of \(x\) is 1, so its degree is 1. B and D are not polynomials, while C has degree 2. Exam tip: check exponents, not coefficients.

Correct answer: C. Never

Explanation: Given p(7)=p(-1), we get 7k-14=-k-14. Thus, 8k=0 and k=0. Then p(x)=-14 is a constant polynomial, not a linear polynomial, because the coefficient of x must be non-zero for a linear polynomial. Therefore, under the given condition, p(x) can never remain linear. Exam tip: In a linear polynomial ax+b, a must satisfy a≠0.

Correct answer: B. 1

Explanation: First simplify the expression: (q(x)-2p(x))=(15x-6)-2(7x+2)=15x-6-14x-4=x-10. The highest power of x is 1, so its degree is 1. A constant polynomial would have degree 0, but here the coefficient of x is 1, not zero. Exam tip: When adding or subtracting linear polynomials, combine like terms first because leading terms may cancel.

Correct answer: B. 12

Explanation: Given \(p(x)=12x-25\), \(p(x+1)=12(x+1)-25=12x-13\). Therefore, \(p(x+1)-p(x)=(12x-13)-(12x-25)=12\). The option \(12x\) is the linear term, not the difference. Exam tip: for a linear polynomial \(ax+b\), \(p(x+1)-p(x)=a\).

Correct answer: C. (\lambda>3)

Explanation: The zero is (-\frac{21}{\lambda}). From (-\frac{21}{\lambda}>-7) and (\lambda>0), we get (\lambda>3).

Correct answer: A. It intersects the \(x\)-axis at exactly one point.

Explanation: Since \(a\ne0\), the graph of \(p(x)=ax+b\) is a non-horizontal straight line. It meets the \(x\)-axis once, at \(x=-\frac{b}{a}\). Exam tip: a zero is the graph’s \(x\)-intercept.

Correct answer: A. \(2x-3r\)

Explanation: \(p(x)+q(x)=(x+5r)+(x-8r)\). Combining like terms gives \(x+x=2x\) and \(5r-8r=-3r\). Therefore, the sum is \(2x-3r\). The option \(2x+13r\) results from incorrectly ignoring the negative sign before \(8r\). Exam tip: When adding polynomials, remove brackets carefully and check the sign of every term.

Correct answer: B. Constant polynomial

Explanation: We get \(p(x)-q(x)=(x+5r)-(x-8r)=x+5r-x+8r=13r\). The \(x\)-terms cancel, so no power of \(x\) remains. Hence, with respect to \(x\), \(13r\) is a constant polynomial. A linear polynomial must have a non-zero coefficient of \(x\). Exam tip: simplify like terms first, then identify the highest power of the variable.

Correct answer: B. -19

Explanation: First evaluate the inner function: \(p(3)=10\times3-29=1\). Then \(p(p(3))=p(1)=10\times1-29=-19\). Therefore, the correct answer is \(-19\). Option 1 is only the value of \(p(3)\); the question requires applying \(p\) once again to that result. Exam tip: In \(p(p(a))\), always evaluate the inner \(p(a)\) first.

Correct answer: C. 35

Explanation: First evaluate the inner function: p(0)=6(0)+5=5. Then p(5)=6(5)+5=35. Therefore, p(p(0))=35. Option 30 results from calculating only 6×5 and forgetting to add the constant term 5. Exam tip: In a composite function, always evaluate the innermost input first.

Correct answer: A. \(n=-\frac{3}{5}\)

Explanation: If \(x=1\) is a zero of \((n+6)x+4n-3\), substituting \(x=1\) must make the polynomial equal to 0. Thus, \((n+6)(1)+4n-3=0\), or \(5n+3=0\). Hence, \(n=-\frac{3}{5}\). For example, \(n=-1\) gives the value \(-2\), not 0, so it is incorrect. Exam tip: substitute the given zero into the polynomial and equate the result to 0.

Correct answer: C. 14

Explanation: Given \(p(x)=\frac{x}{8}+a\), substitute \(x=24\): \(p(24)=\frac{24}{8}+a=3+a\). Since \(p(24)=17\), we get \(3+a=17\), so \(a=14\). Option 17 is the given value of \(p(24)\), not the value of \(a\). Exam tip: When a polynomial value is given, first substitute the specified value of \(x\) into the expression.

Correct answer: A. -10

Explanation: First add the polynomials: \(p(x)+q(x)=7x+a+4x-12=11x+a-12\). Since 2 is a zero, put \(x=2\): \(22+a-12=0\). Thus \(a+10=0\), so \(a=-10\). If \(a=0\), the value of the polynomial at \(x=2\) would be 10, not zero. Exam tip: when a zero is given, substitute that value in the polynomial and set the result equal to 0.

Correct answer: A. \(c=-7\)

Explanation: If \(-4\) is a zero of \(p(x)=(c-7)x+8c\), then \(p(-4)=0\). Thus, \(-4(c-7)+8c=0\). On simplifying, \(-4c+28+8c=0\), so \(4c+28=0\), which gives \(c=-7\). For \(c=7\), the constant term is \(56\), so \(-4\) cannot be a zero. Exam tip: for a given zero \(a\), always substitute it and use \(p(a)=0\).

Correct answer: C. No value of \(c\)

Explanation: If \(-6\) is a zero of \(p(x)\), then \(p(-6)=0\) must hold. However, \(p(-6)=(c-3)(-6)+6c=-6c+18+6c=18\), which can never be \(0\). Hence, no value of \(c\) makes \(-6\) a zero of this polynomial. Exam tip: To test a given zero \(a\), substitute it directly and check whether \(p(a)=0\).