Graph polynomials and read end behavior, do polynomial arithmetic, prove identities, divide, find real and complex zeros, apply the root theorems, and transform parent graphs — with the HP Prime as your checker.
This chapter uses the HP Prime two ways: the Function app to graph, table, and read features straight off a curve, and the CAS to expand, factor, divide, and find roots symbolically. For lessons where a CAS command answers the problem (factoring, finding zeros, finding roots, dividing), treat it as a checker — show your algebra, then confirm. The ✅ Quick-verify recipe under each lesson is always fair game.
Problem types: Sketch a polynomial from a table of values; find zeros and turning points; predict end behavior from the leading coefficient and degree.
The Function app graphs the polynomial and builds the table for you, so you can see the zeros, the turning points, and which way each end heads — then check the end behavior you reasoned out from the leading term.
Path
Apps ▸ Function ▸ Symb ▸ F1(X)= ▸ Plot ▸ (Menu ▸ Fcn ▸ Root or Extremum); table via Num
📝 Worked example
Problem: Graph f(x) = −0.5x⁴ + 3x² + 2 and describe its end behavior.
Do this on the calculator:
AppsOpens the App Library — the screen with all the app icons.
FunctionHighlight the Function app icon and press Enter to open it. This is the grapher; it starts on the Symb (symbolic/equation) screen.
Type-0.5*X^4+3*X^2+2With F1(X) highlighted, type the function. Use the [X,θ,T,N] key for X and the x^y key for the powers.
EnterLocks in the F1(X) definition.
NumPress the Num key to see the table of values — at X=−2,−1,0,1,2 it reads 6, 4.5, 2, 4.5, 6.
PlotPress the Plot key to draw the graph — you'll see an M-shaped curve (two peaks, one valley).
trace near the left bumpUse the arrow keys to move the blinking tracer over toward the left peak.
Menu▸Fcn▸ExtremumTap the Menu soft button, then Fcn, then Extremum to find that turning point and print its coordinates at the bottom of the screen.
You'll see: An M-shaped curve (two maxes, one min). The Num table at X=−2,−1,0,1,2 reads 6, 4.5, 2, 4.5, 6. Extremum prints the turning-point coordinates at the bottom of Plot view. Leading term −0.5x⁴ (negative coefficient, even degree) → both ends point down.
✅ Quick-verify (check your hand-work)
NumPress the Num key to open the table, then put the cursor in the X column.
Type100Type a big value and press Enter — the table jumps to the X=100 row and shows F1 is a large negative number.
Type-100Now type a big negative value (use the (−) negative-sign key) and press Enter — F1 is again a large negative number. Both ends down, matching 'negative leading coefficient, even degree.'
#3.2—Adding, Subtracting, and Multiplying Polynomials
Problem types: Add, subtract, and multiply polynomials; collect like terms; expand products.
🎣 Method (teach-the-tool)
Tool
CAS expand() and simplify()
Why this tool
expand multiplies everything out and collects like terms in one step, so it's the fastest way to confirm a product or a sum you did by hand. These are CHECK tools — you still show the distributive-property / box-method work.
Path
CAS ▸ Toolbox ▸ CAS ▸ (type or pick) expand( ) ▸ Enter
📝 Worked example
Problem: Multiply (2m + 5)(3m² − 4m + 2).
Do this on the calculator:
CASPress the CAS key to switch to CAS view — the symbolic screen where the calculator does algebra exactly.
Toolbox▸CASPress the Toolbox key to open the menus, then tap the CAS tab to find the algebra commands.
Typeexpand((2*m+5)*(3*m^2-4*m+2))Pick or type expand, then fill in the two factors. Use the x^y key for the power.
EnterThe screen returns 6*m^3+7*m^2-16*m+10 — exactly your boxed hand answer.
You'll see: expand((2*m+5)*(3*m^2-4*m+2)) returns 6*m^3+7*m^2-16*m+10 — exactly the boxed hand answer.
✅ Quick-verify (check your hand-work)
Typeexpand((6*m^3+7*m^2-16*m+10) - (your answer))In CAS view, type expand of (the calculator's result minus your hand answer).
EnterThe screen prints 0 — the difference vanished, so your expansion is correct.
Problem types: Use identities (difference of squares, square of a sum, sum/difference of cubes) to multiply and factor; verify an identity holds.
🎣 Method (teach-the-tool)
Tool
CAS expand() (both sides) and factor()
Why this tool
An identity says two expressions are equal for all values. expand turns the left side into the right; factor goes the other way. Either direction lets you confirm the identity you applied by hand is the right one.
Path
CAS ▸ Toolbox ▸ CAS ▸ expand( ) or factor( ) ▸ Enter
📝 Worked example
Problem: Factor 9m⁴ − 25n⁶ (difference of squares).
Do this on the calculator:
CASPress the CAS key to switch to CAS view — the symbolic algebra screen.
Toolbox▸CASPress the Toolbox key, then tap the CAS tab to reach the algebra commands.
Typefactor(9*m^4-25*n^6)Pick or type factor, then enter the expression. Use the x^y key for the powers.
EnterThe screen returns (3*m^2-5*n^3)*(3*m^2+5*n^3) — your boxed answer.
You'll see: factor(9*m^4-25*n^6) returns (3*m^2-5*n^3)*(3*m^2+5*n^3) — the boxed answer. (Going the other way, expand((2*X^2+y^3)^2) returns 4*X^4+4*X^2*y^3+y^6.)
✅ Quick-verify (check your hand-work)
Typesimplify((2*X^2+y^3)^2 - (4*X^4+4*X^2*y^3+y^6))In CAS view, type simplify of (one side minus the other side) of the identity you're checking.
EnterThe screen prints 0 — the two sides are equal, so the identity holds.
Problem types: Divide polynomials by long division or synthetic division; write P(x) as quotient plus remainder over divisor; use the result to factor.
🎣 Method (teach-the-tool)
Tool
CAS quorem() (quotient + remainder) — and propfrac() for the quotient-plus-remainder form
Why this tool
quorem returns the quotient and remainder together as a list, which is exactly what long or synthetic division produces — so you can check both at once. propfrac rewrites the whole rational expression in the 'quotient + remainder/divisor' shape the notes ask for.
Path
CAS ▸ Toolbox ▸ CAS ▸ quorem( , ) ▸ Enter (or propfrac( (...)/(...) ))
📝 Worked example
Problem: Divide P(x) = x³ + 5x² + 6x + 9 by x + 3.
Do this on the calculator:
CASPress the CAS key to switch to CAS view — the symbolic algebra screen.
Toolbox▸CASPress the Toolbox key, then tap the CAS tab to reach the algebra commands.
Typequorem(X^3+5*X^2+6*X+9, X+3)Pick or type quorem, then enter the dividend, a comma, and the divisor. Use the x^y key for the powers.
EnterThe screen returns [X^2+2*X, 9] — the quotient and the remainder, i.e. x²+2x + 9/(x+3).
You'll see: quorem(X^3+5*X^2+6*X+9, X+3) returns [X^2+2*X, 9] (quotient, remainder) — i.e. x²+2x + 9/(x+3), the boxed answer.
✅ Quick-verify (check your hand-work)
Typeexpand((X+3)*(X^2+2*X) + 9)In CAS view, type expand of (divisor × quotient + remainder) to rebuild the original.
EnterThe screen prints X^3+5*X^2+6*X+9 — the original P(x), so the division checks out.
Problem types: Find zeros by factoring or synthetic division; use multiplicity to decide cross vs. bounce; find real and complex zeros; solve polynomial inequalities by graphing.
🎣 Method (teach-the-tool)
Tool
CAS factor() / zeros(); Function app Fcn ▸ Root for real zeros
Why this tool
factor and zeros hand you every zero (and, with cZeros, the complex ones) directly. For just the real x-intercepts, the Function app's Root tool reads them off the graph — useful for the inequality problems where you need to see where the curve is above or below the axis.
Path
CAS ▸ Toolbox ▸ CAS ▸ factor( ) or zeros( ) ▸ Enter; real roots: Apps ▸ Function ▸ Plot ▸ Menu ▸ Fcn ▸ Root
📝 Worked example
Problem: Find the zeros of f(x) = 4x³ + 4x² − 24x.
Do this on the calculator:
CASPress the CAS key to switch to CAS view — the symbolic algebra screen.
Toolbox▸CASPress the Toolbox key, then tap the CAS tab to reach the algebra commands.
Typezeros(4*X^3+4*X^2-24*X)Pick or type zeros, then enter the polynomial. Use the x^y key for the powers.
EnterThe screen returns [−3, 0, 2] — every zero at once.
You'll see: zeros(4*X^3+4*X^2-24*X) returns [−3, 0, 2]. (For f(x)=x³+x²−3x−6, factor(X^3+X^2-3*X-6) returns (X-2)*(X^2+3*X+3) — real zero X=2 plus the quadratic.)
✅ Quick-verify (check your hand-work)
HomePress the Home key for plain number-crunching.
Type2Type the candidate zero you want to test (here, 2).
Sto▶▸XTap the [Sto▶] (store-into-variable) softkey, then press X to save 2 into the variable X.
EnterConfirms the store — now X holds 2.
TypeX^3+X^2-3*X-6Type the polynomial f(x) you're checking.
EnterThe screen prints 0 — f(2)=0, so X=2 is a zero.
Problem types: List possible rational roots; find all real and complex roots; use conjugate-root theorems to write a polynomial from given roots.
🎣 Method (teach-the-tool)
Tool
CAS proot() (all roots incl. complex) and factor(); Home evaluation to test rational-root candidates
Why this tool
proot returns every root of the polynomial — rational, irrational, and complex — in one shot, so you can confirm the roots you found by synthetic division and the quadratic formula. To test a single Rational-Root-Theorem candidate, just evaluate f at it in Home.
Path
CAS ▸ Toolbox ▸ CAS ▸ proot( ) or factor( ) ▸ Enter; candidate test: Home ▸ store value ▸ evaluate f
📝 Worked example
Problem: Find all four roots of 3x⁴ + 4x³ + 2x² − x − 2 = 0 (you found −1 and 2/3 by synthetic division; the last two come from the quadratic formula).
Do this on the calculator:
CASPress the CAS key to switch to CAS view — the symbolic algebra screen.
Toolbox▸CASPress the Toolbox key, then tap the CAS tab to reach the algebra commands.
Typeproot([3,4,2,-1,-2])Pick or type proot, then enter the coefficients in square brackets, highest power first (3, 4, 2, −1, −2).
EnterThe screen returns all four roots as a vector — the two real ones (−1 and ≈0.6667) and the two complex conjugate roots (−1±i√3)/2.
You'll see: proot([3,4,2,-1,-2]) returns all four roots as a vector — the two real roots (−1 and ≈0.6667) and the two complex conjugate roots (−1±i√3)/2. (You may also pass the polynomial: proot(3*X^4+4*X^3+2*X^2-X-2).)
✅ Quick-verify (check your hand-work)
HomePress the Home key for plain number-crunching.
Type2/3Type the Rational-Root candidate you want to test (here, 2/3).
Sto▶▸XTap the [Sto▶] (store-into-variable) softkey, then press X to save 2/3 into the variable X.
EnterConfirms the store — now X holds 2/3.
Type3*X^4+4*X^3+2*X^2-X-2Type the polynomial f(x) you're testing.
EnterThe screen prints 0 — f(2/3)=0, so 2/3 is a root.
Problem types: Classify functions as even/odd; identify stretches, reflections, and shifts on cubic and quartic parents; write a transformed equation from a graph.
🎣 Method (teach-the-tool)
Tool
Function app — overlay the parent and the transformed function
Why this tool
Graphing the parent (X³ or X⁴) and the transformed version on the same axes lets you SEE every stretch, reflection, and shift, and confirm the transformation equation you wrote produces exactly that curve.
Path
Apps ▸ Function ▸ Symb ▸ F1(X)=parent, F2(X)=transformed ▸ Plot
📝 Worked example
Problem: Describe how g(x) = 3x⁴ − 18 transforms the parent x⁴.
Do this on the calculator:
AppsOpens the App Library — the screen with all the app icons.
FunctionHighlight the Function app icon and press Enter to open it. It starts on the Symb (symbolic/equation) screen.
TypeX^4With F1(X) highlighted, type the parent function. Use the x^y key for the power.
EnterLocks in F1(X)=X^4, the parent.
Type3*X^4-18Move down to F2(X) and type the transformed function.
EnterLocks in F2(X)=3*X^4-18.
PlotPress the Plot key to draw both curves on the same axes — F2 is F1 stretched vertically by 3 and shifted down 18.
You'll see: Two overlaid curves: F2 is F1 stretched vertically by 3 and shifted down 18. (For h(x)=−5(X−2)³+4 use F1(X)=X^3, F2(X)=-5*(X-2)^3+4 → stretch 5, reflect, right 2, up 4.)
✅ Quick-verify (check your hand-work)
CASPress the CAS key to switch to CAS view — the symbolic algebra screen.
Typesimplify(f(-X) - f(X))Type simplify of (your function with −X minus your function with X) to test for even symmetry.
EnterThe screen prints 0 — the two expressions are equal, so the function is even (and your transformation matches).